Finite Group DatabasebetaFinite Group Databasebeta| order | 12810 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C3×C7⋊(D61⋊C5) |
| unambiguous semidirect product | (C3×D427)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C7⋊(D61⋊C5) |
| commutator, cosocle, hypocenter | C7×C61 | C2×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C3×C7×C61 | C2×C5 |
| LCS | G ▷ C7×C61 | |
| UCS | C1 ◁ C3 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 1281 |
A minimal faithful permutation representation — 2 generators on 71 points:
(2,6,7,11,28,3,4,12,14,22)(5,19,20,29,44,9,10,15,37,27)(8,31,32,33,45,16,17,36,49,50)(13,18,43,51,48,25,26,34,58,39)(21,35,56,40,41,24,38,52,23,57)(30,59,53,42,60,46,47,61,55,54)(63,64)(65,66)(67,68)(69,70,71);(1,2,5,11,23,42,29,44,25,4,8,18,35,53,56,47,60,32,10,21,7,15,30,43,58,57,50,22,39,17,33,49,31,48,28,45,41,51,34,46,20,12,24,19,36,54,59,52,61,38,26,16,6,13,27,37,55,40,14,9,3)(62,63,65,67,68,66,64)201 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12810, 28);