Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C24 ⋊ GL(3,𝔽2) |
| derived | AGL(3,𝔽2) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | AGL(3,𝔽2) | C2 |
| Frattini, socle, cosocle | C23 | C2×GL(3,𝔽2) |
| Fitting | C24 | GL(3,𝔽2) |
| LCS | G ▷ AGL(3,𝔽2) | |
| UCS | C1 | |
119 conjugacy classes of subgroups; only the 5 normal subgroups are shown.
This group has 119 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) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,4,7)(5,11,8)(6,9,12)(10,15,13);(1,5,12,14,9,15,2,13)(3,11,7,8,4,10,6,16)16 conjugacy classes — view the full character table.
| maximal subgroup of | #72227517 |
| maximal quotient of | #72227541 |
| automorphism group of | AGL(3,𝔽2) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,4,7)(5,11,8)(6,9,12)(10,15,13), (1,5,12,14,9,15,2,13)(3,11,7,8,4,10,6,16));