Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×C4×#5933 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | #1020 |
| commutator, hypocenter | #997 | C22×C4 |
| Frattini | C22 | C22×#1020 |
| Fitting | C2×C4×H(5,𝔽2) | D3 |
| socle | C23 | C2×#1020 |
| prime radical | C22×#997 | C22 |
| cosocle | C2×#997 | C23 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22×C4 | |
3121 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 3121 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 128 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2)(9,20,16)(10,19,17)(12,18,15)(13,22,21);(3,4,5,6)(7,16,11,19)(8,15,14,22)(9,20,17,10)(12,18,21,13);(7,8,11,14)(9,15,17,22)(10,18,20,13)(12,19,21,16)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72245733, C2×#72245388, C2×#72245725, … |
| maximal normal subgroup of | C2×#72245733, C2×#72245388, C2×#72245725, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(9,20,16)(10,19,17)(12,18,15)(13,22,21), (3,4,5,6)(7,16,11,19)(8,15,14,22)(9,20,17,10)(12,18,21,13), (7,8,11,14)(9,15,17,22)(10,18,20,13)(12,19,21,16));