Finite Group DatabasebetaFinite Group Databasebeta| order | 7056 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C7 ⋊ C3×SO(3,𝔽7) |
| socle, derived | C7×GL(3,𝔽2) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | C7×GL(3,𝔽2) | C2×C3 |
| Fitting | C7 | C3×SO(3,𝔽7) |
| LCS | G ▷ C7×GL(3,𝔽2) | |
| UCS | C1 | |
114 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 114 conjugacy classes of subgroups — too many to draw. Only its 8 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,3)(2,4)(5,6)(8,10,9,11,15,14,13,12);(1,4,3)(2,5,7)(9,13,12)(10,15,14)33 conjugacy classes — view the full character table.
| maximal subgroup of | #72244493 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3)(2,4)(5,6)(8,10,9,11,15,14,13,12), (1,4,3)(2,5,7)(9,13,12)(10,15,14));