Finite Group DatabasebetaFinite Group Databasebeta| order | 56448 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
No aliases for this group.
| socle, derived | GL(3,𝔽2)2 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | GL(3,𝔽2)2 | C2 |
| LCS | G ▷ GL(3,𝔽2)2 | |
| UCS | C1 | |
413 conjugacy classes of subgroups; only the 5 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 5 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,8,6,3,7,2)(9,10,13,11,16,14,12,15);(1,6,8)(4,7,5)(9,11,10,12)(13,14,15,16)42 conjugacy classes — view the full character table.
| holomorph of | GL(3,𝔽2) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,8,6,3,7,2)(9,10,13,11,16,14,12,15), (1,6,8)(4,7,5)(9,11,10,12)(13,14,15,16));