Finite Group DatabasebetaFinite Group Databasebeta| order | 12810 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊#14579424 |
| 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:
(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,34)(18,26,31)(21,29,28)(23,32,25)(27,37,44)(30,39,41)(33,42,38)(35,45,36)(40,48,52)(43,50,49)(46,53,47)(51,56,57)(54,58,55)(59,61,60)(63,66,67,64,65,68)(70,71)(72,73);(1,2,4,8,15,24,35,46,54,59,58,42,20,5,9,16,25,36,37,14,23,33,43,51,53,32,12,19,28,38,47,48,26,7,13,21,30,40,45,22,31,41,49,55,56,39,17,3,6,11,18,27,34,44,52,57,60,61,50,29,10)(62,63,65,68,67,66,64)(69,70,72,73,71)Polycyclic presentation:
2,3,5,7,61|c^a=c^4,d^a=d^6,d^b=d^2,e^b=e^47397 conjugacy classes — view the full character table.
| maximal subgroup of | #14579420⋊D5, #14579428⋊D5, D5×#14579424, … |
| maximal normal subgroup of | #14579420⋊D5, #14579428⋊D5, D5×#14579424, … |
| maximal quotient of | #38784982, C2×C5⋊#14579424, C5⋊#21440254, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12810, 55);