Finite Group DatabasebetaFinite Group Databasebeta| order | 12810 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #14579425⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C61 | C2×C3×C5 |
| LCS | G ▷ C7×C61 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 427 |
A minimal faithful permutation representation — 2 generators on 68 points:
(2,7,31,20,19,6,4,16,51,39,3,14,9,34,38)(5,26,53,49,43,25,12,46,54,59,11,45,28,60,61)(8,36,52,29,27,15,18,50,30,33,17,32,37,13,47)(10,24,41,55,48,42,23,44,58,57,22,21,35,56,40)(63,66,67,64,65,68);(1,2,5,13,18,32,41,59,22,4,10,24,38,3,8,19,16,35,12,29,9,21,34,54,61,31,11,27,48,23,43,55,46,25,39,57,37,56,44,30,28,50,51,42,60,52,47,6,15,33,53,26,36,45,58,49,7,14,20,40,17)(62,63,65,68,67,66,64)Polycyclic presentation:
2,3,5,7,61|d^a=d^6,d^b=d^2,e^b=e^13,e^c=e^955 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12810, 68);