Finite Group DatabasebetaFinite Group Databasebeta| order | 8064 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| direct product | S4×SL(2,𝔽7) |
| derived | A4×SL(2,𝔽7) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | S4×GL(3,𝔽2) |
| commutator, hypocenter | A4×SL(2,𝔽7) | C2 |
| Fitting, socle | C23 | D3×GL(3,𝔽2) |
| cosocle | C2×A4 | C2×GL(3,𝔽2) |
| LCS | G ▷ A4×SL(2,𝔽7) | |
| UCS | C1 ◁ C2 | |
406 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,4)(5,19,17,9,7,18,14,6)(8,15,13,16,12,11,10,20);(3,4)(5,19,13,12,7,18,10,8)(6,20,14,11,9,16,17,15)55 conjugacy classes — view the full character table.
| maximal subgroup of | GL(3,𝔽2)×SL(2,𝔽7) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4)(5,19,17,9,7,18,14,6)(8,15,13,16,12,11,10,20), (3,4)(5,19,13,12,7,18,10,8)(6,20,14,11,9,16,17,15));