Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | C4×#10621462 |
| derived | SL(2,𝔽7) ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2×C4 | SO(3,𝔽7) |
| commutator, hypocenter | SL(2,𝔽7) | C2×C4 |
| Frattini, socle | C22 | C2×SO(3,𝔽7) |
| prime radical | C2×#10621462 | C2 |
| cosocle | C2×SL(2,𝔽7) | C22 |
| LCS | G ▷ SL(2,𝔽7) | |
| UCS | C1 ◁ C2×C4 | |
190 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 16 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) = 64 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,3,2,4)(5,17,12,11,8,7,20)(6,16,14,10,19,9,18);(1,2)(3,4)(5,20,6,10,8,13)(7,14,15,9,19,12)(17,18)64 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C4×#10621462 |
| maximal normal subgroup of | C3×C4×#10621462 |
| maximal quotient of | C3×C4×#10621462 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,2,4)(5,17,12,11,8,7,20)(6,16,14,10,19,9,18), (1,2)(3,4)(5,20,6,10,8,13)(7,14,15,9,19,12)(17,18));