Finite Group DatabasebetaFinite Group Databasebeta| order | 12705 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C7⋊((C112⋊C3)⋊C5) |
| unambiguous semidirect product | #10597243⋊ΓL(1,𝔽8) |
| unambiguous semidirect product | (C7⋊(C112⋊C3))⋊C5 |
| Fitting, socle, derived, Hall | C7×C112 ⋊ C3×C5 |
| Hall | C7×#10597243 ⋊ C3 |
| Hall | C7⋊(C112⋊C3) ⋊ C5 |
| Hall | #10597243 ⋊ ΓL(1,𝔽8) |
| Sylow | C112 ⋊ C5×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ (C112⋊C3)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C112 | C3×C5 |
| LCS | G ▷ C7×C112 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 231 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,4)(3,6,29,31,28,16,8,14,22,21,7,15,18,17,25)(5,26,23,30,10,20,13,11,32,19,12,33,27,24,9)(35,36,37)(38,40,39);(1,3,9,21,8,18,32,23,20,31,33)(2,6,5,7,14,17,19,30,13,28,27)(4,24,16,11,10,12,25,22,15,26,29)(34,35,36,38,37,39,40)Polycyclic presentation:
3,5,7,11^2|c^a=c^2,d^a=d^2e^10,e^a=d^7e^8,d^b=d^4,e^b=e^481 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12705, 11);