Finite Group DatabasebetaFinite Group Databasebeta| order | 2352 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C7 ⋊ SO(3,𝔽7) |
| socle, derived | C7×GL(3,𝔽2) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | C7×GL(3,𝔽2) | C2 |
| Fitting | C7 | SO(3,𝔽7) |
| LCS | G ▷ C7×GL(3,𝔽2) | |
| UCS | C1 | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,6)(2,7)(3,5)(8,9,13,11,14,15,10,12);(1,3,5,6,2,4,7)(8,13,15,9,14,10,12)27 conjugacy classes — view the full character table.
| maximal subgroup of | #72244479 |
| maximal normal subgroup of | #72244479 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,6)(2,7)(3,5)(8,9,13,11,14,15,10,12), (1,3,5,6,2,4,7)(8,13,15,9,14,10,12));