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  "body": "\n## Example\n\nUp to isomorphism, the [[groups|G5S1M9]] of order $n \\le 15$ are the following.\nHere $C_n$ is the [[cyclic group|0KVGH0J5B3]] of order $n$ and $\\times$ is the [[direct product|00HF4RJ5QN]].\nThe group $D_{2m}$ is the [[dihedral group|0JZ27DMJ5D]] of order $2m$, and $Q_8$ is the [[quaternion group|0NRJGW864D]].\nThe group $A_4$ is the [[alternating group|0Z5905Q73D]] on $4$ letters, and $\\mathrm{Dic}_{12} = C_3 \\rtimes C_4$ is the dicyclic group of order $12$.\n\n1. Orders $1$, $2$, $3$, $5$, $7$, $11$, $13$: only $C_n$.\n2. Order $4$: $C_4$ and the [[Klein four-group|0PMRQJVTME]] $C_2 \\times C_2$.\n3. Order $6$: $C_6$ and $D_6 \\cong S_3$, the [[symmetric group|0515A6D2Y1]].\n4. Order $8$: $C_8$, $C_4 \\times C_2$, $C_2 \\times C_2 \\times C_2$, $D_8$ and $Q_8$.\n5. Order $9$: $C_9$ and $C_3 \\times C_3$.\n6. Order $10$: $C_{10}$ and $D_{10}$.\n7. Order $12$: $C_{12}$, $C_6 \\times C_2$, $A_4$, $D_{12}$ and $\\mathrm{Dic}_{12}$.\n8. Order $14$: $C_{14}$ and $D_{14}$.\n9. Order $15$: only $C_{15}$.\n\nSo the numbers of groups of orders $1, 2, \\dots, 15$ are $1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1$.\nFor order $16$ there are already $14$ groups.\n\n## Remarks\n\nGroups of prime order $p$ are cyclic, which explains the first line.\nFor primes $p \u003c q$, a group of order $pq$ is cyclic unless $p$ divides $q - 1$.\nThis explains why $15 = 3 \\cdot 5$ gives only $C_{15}$, while $6$, $10$ and $14$ also give a dihedral group.\nA group of order $p^2$ is abelian, so it is $C_{p^2}$ or $C_p \\times C_p$, as for orders $4$ and $9$.\nIn each order the abelian groups are given by the [[classification of finite abelian groups|07CQR9BW3R]].\n\n## Explanations\n\n### L1 human\n\nThis is the start of a catalogue of all finite groups, sorted by size.\nSmall sizes allow few groups: a prime number of elements forces a cyclic group.\nThe first interesting sizes are $8$ and $12$, which already show non-abelian groups such as the quaternion group and the alternating group $A_4$.\n",
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