Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×C3×SO(3,𝔽7) |
| socle | C2×C3×GL(3,𝔽2) ⋊ C2 |
| derived | GL(3,𝔽2) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2×C3 | SO(3,𝔽7) |
| commutator, cosocle, hypocenter | GL(3,𝔽2) | C22×C3 |
| Frattini | C1 | G |
| socle | C2×C3×GL(3,𝔽2) | C2 |
| LCS | G ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C2×C3 | |
157 conjugacy classes of subgroups; only the 14 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 14 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2)(6,11,10,12)(7,9,8,13);(3,5,4)(6,8,13,11,10,7)54 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×SO(3,𝔽7), C3×C4×SO(3,𝔽7) |
| maximal normal subgroup of | C3×C4×SO(3,𝔽7) |
| maximal quotient of | C2×C3×#10621462, C3×D3×SO(3,𝔽7), C3×C4×SO(3,𝔽7), … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(6,11,10,12)(7,9,8,13), (3,5,4)(6,8,13,11,10,7));