Finite Group DatabasebetaFinite Group Databasebeta| order | 12810 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊((C61⋊C3)⋊D7) |
| unambiguous semidirect product | (C61⋊C3)⋊D35 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5×C7×C61 | C2×C3 |
| LCS | G ▷ C5×C7×C61 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 2135 |
A minimal faithful permutation representation — 2 generators on 73 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,40)(12,26,22,20,16,13)(19,45,46,30,31,56)(21,43,44,32,29,37)(23,25,42,34,35,38)(33,58,59,48,47,54)(36,41,57,50,49,55)(39,61,60,52,53,51)(63,64)(65,66)(67,68)(70,71)(72,73);(1,2,5,11,19,31,47,53,55,56,50,34,22,7,6,12,21,33,49,18,29,38,40,54,60,44,15,26,14,25,39,52,35,10,16,24,37,51,59,28,42,43,27,41,48,32,20,4,8,13,23,36,46,57,61,58,45,30,17,9,3)(62,63,65,67,68,66,64)(69,70,72,73,71)Polycyclic presentation:
2,3,5,7,61|c^a=c^4,d^a=d^6,e^a=e^60,e^b=e^47407 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12810, 57);