Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| isoclinism | isoclinic class of GL(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×C4×GL(3,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C3×C4 | GL(3,𝔽2) |
| commutator, hypocenter | GL(3,𝔽2) | C3×C4 |
| Frattini, cosocle | C2 | C2×C3×GL(3,𝔽2) |
| socle, prime radical | C2×C3×GL(3,𝔽2) | C2 |
| LCS | G ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C3×C4 | |
147 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 147 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2,3)(4,5,6,7)(8,9,11)(10,12,14);(8,12,14,11,9,13,10)72 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,5,6,7)(8,9,11)(10,12,14), (8,12,14,11,9,13,10));