Finite Group DatabasebetaFinite Group Databasebeta| order | 12810 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊#14579421 |
| 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)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,32,30,31,48)(21,44,45,33,29,38)(23,25,43,35,36,39)(34,59,53,50,47,57)(37,42,49,51,52,58)(40,60,61,55,56,54)(63,64,65)(66,68,67)(70,71)(72,73);(1,2,5,11,19,32,49,60,59,46,50,33,20,4,8,13,23,37,53,28,43,44,27,42,55,36,10,16,24,38,54,61,45,15,26,14,25,40,52,18,29,39,41,57,51,35,22,7,6,12,21,34,31,47,56,58,48,30,17,9,3)(62,63,64,66,65,67,68)(69,70,72,73,71)Polycyclic presentation:
2,3,5,7,61|c^a=c^4,e^a=e^60,d^b=d^2,e^b=e^13370 conjugacy classes — view the full character table.
| maximal subgroup of | #38785062, #38785064, #38785074, … |
| maximal normal subgroup of | #38785062, #38785064, #38785074, … |
| maximal quotient of | #38784979, C2×C5⋊#14579421, C5⋊#21440251, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12810, 52);