Finite Group DatabasebetaFinite Group Databasebeta| order | 12642 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C72×C43 ⋊ C2×C3 |
| Hall | #20263480 ⋊ C2 |
| Hall | C7×D301 ⋊ C3 |
| Sylow | C72 ⋊ D43⋊C3 |
| Sylow | C43 ⋊ #64423 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C43 | C2×C3 |
| LCS | G ▷ C72×C43 | |
| UCS | C1 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 602 |
A minimal faithful permutation representation — 2 generators on 57 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,37)(12,26,22,20,16,13)(19,29,34,21,30,40)(23,25,39,31,32,35)(33,38,42,36,41,43)(45,48,49,46,47,50)(52,53,54)(55,57,56);(1,2,5,11,19,20,4,8,13,23,33,32,10,16,24,34,42,41,18,29,35,37,28,39,30,27,38,43,40,15,26,14,25,36,31,22,7,6,12,21,17,9,3)(44,45,47,50,49,48,46)(51,52,53,55,54,56,57)Polycyclic presentation:
2,3,7^2,43|c^a=c^6,e^a=e^42,c^b=c^2,d^b=d^2,e^b=e^6360 conjugacy classes — view the full character table.
| maximal subgroup of | #38760453, #38760460, #38760462, … |
| maximal normal subgroup of | #38760453, #38760460, #38760462, … |
| maximal quotient of | #38760322, C2×#27162415, #49181565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12642, 80);