Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | D3×SO(3,𝔽7) |
| Fitting | C3 ⋊ C2×SO(3,𝔽7) |
| socle, derived | C3×GL(3,𝔽2) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | C3×GL(3,𝔽2) | C22 |
| Fitting | C3 | C2×SO(3,𝔽7) |
| LCS | G ▷ C3×GL(3,𝔽2) | |
| UCS | C1 | |
149 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,3,2)(4,11,10,9,8,5);(2,3)(4,8,7,11,9,5,6,10)27 conjugacy classes — view the full character table.
| maximal subgroup of | #72244493, C3×D3×SO(3,𝔽7) |
| maximal normal subgroup of | C3×D3×SO(3,𝔽7) |
| maximal quotient of | D3×#10621462, C3×D3×SO(3,𝔽7) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,2)(4,11,10,9,8,5), (2,3)(4,8,7,11,9,5,6,10));