Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#11714271 |
| derived | #5464 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #85559 |
| commutator, hypocenter | #5464 | C2×C4 |
| Frattini | C23 | C2×#1020 |
| Fitting | C22×H(3,𝔽4) | D3 |
| socle | C24 | #1020 |
| prime radical | C22×#5464 | C2 |
| cosocle | C2×#5464 | C22 |
| LCS | G ▷ #5464 | |
| UCS | C1 ◁ C22 | |
1706 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 1706 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 22 points:
(3,4,5,6)(7,10,21,16,8,13,22,19)(9,18,17,14,12,15,20,11);(1,2)(8,9,12)(10,14,21)(11,17,18)(13,19,20)(15,16,22)48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72310668, C2×C4×#85559, C2×#72310694, … |
| maximal normal subgroup of | C2×#72310668, C2×C4×#85559, C2×#72310694, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,5,6)(7,10,21,16,8,13,22,19)(9,18,17,14,12,15,20,11), (1,2)(8,9,12)(10,14,21)(11,17,18)(13,19,20)(15,16,22));