Element structure of groups of order 24
This article gives specific information, namely, element structure, about a family of groups, namely: groups of order 24.
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Conjugacy class structure
FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:
Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group
Full listing
| Group | Second part of GAP ID (ID is (24,second part)) | Nilpotency class | Derived length | Number of conjugacy classes of size 1 (= order of center) | Number of conjugacy classes of size 2 | Number of conjugacy classes of size 3 | Number of conjugacy classes of size 4 | Number of conjugacy classes of size 6 | Number of conjugacy classes of size 8 | Total number of conjugacy classes |
|---|---|---|---|---|---|---|---|---|---|---|
| nontrivial semidirect product of Z3 and Z8 | 1 | not nilpotent | 2 | 4 | 4 | 4 | 0 | 0 | 0 | 12 |
| cyclic group:Z24 | 2 | 1 | 1 | 24 | 0 | 0 | 0 | 0 | 0 | 24 |
| special linear group:SL(2,3) | 3 | not nilpotent | 3 | 2 | 0 | 0 | 4 | 1 | 0 | 7 |
| dicyclic group:Dic24 | 4 | not nilpotent | 2 | 2 | 5 | 0 | 0 | 2 | 0 | 9 |
| direct product of S3 and Z4 | 5 | not nilpotent | 2 | 4 | 4 | 4 | 0 | 0 | 0 | 12 |
| dihedral group:D24 | 6 | not nilpotent | 2 | 2 | 5 | 0 | 0 | 2 | 0 | 9 |
| direct product of Dic12 and Z2 | 7 | not nilpotent | 2 | 4 | 4 | 4 | 0 | 0 | 0 | 12 |
| SmallGroup(24,8) | 8 | not nilpotent | 2 | 2 | 5 | 0 | 0 | 2 | 0 | 9 |
| direct product of Z6 and Z4 (also, direct product of Z12 and Z2) | 9 | 1 | 1 | 24 | 0 | 0 | 0 | 0 | 0 | 24 |
| direct product of D8 and Z3 | 10 | 2 | 2 | 6 | 9 | 0 | 0 | 0 | 0 | 15 |
| direct product of Q8 and Z3 | 11 | 2 | 2 | 6 | 9 | 0 | 0 | 0 | 0 | 15 |
| symmetric group:S4 | 12 | not nilpotent | 3 | 1 | 0 | 1 | 0 | 2 | 1 | 5 |
| direct product of A4 and Z2 | 13 | not nilpotent | 2 | 2 | 0 | 2 | 4 | 0 | 0 | 8 |
| direct product of D12 and Z2 (also direct product of S3 and V4) | 14 | not nilpotent | 2 | 4 | 4 | 4 | 0 | 0 | 0 | 12 |
| direct product of E8 and Z3 | 15 | 1 | 1 | 24 | 0 | 0 | 0 | 0 | 0 | 24 |