Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | C2×#11711634 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | #1020⋊C2 |
| commutator, cosocle | C2×#997 | C23 |
| Frattini | C24 | C22×S4 |
| Fitting | C22×B(4,𝔽2) | D3 |
| prime radical | C22×B(4,𝔽2)⋊C3 | C2 |
| hypocenter | #997 | C2×D4 |
| LCS | G ▷ C2×#997 ▷ #997 | |
| UCS | C1 ◁ C23 | |
1922 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 1922 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 64 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2)(7,17,10)(8,15,13)(11,20,19)(14,22,18);(4,6)(9,16)(10,19)(12,21)(13,18)(15,22)(17,20);(3,4)(5,6)(7,8,11,14)(9,15,16,22)(10,18,19,13)(12,20,21,17)62 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72241395, C2×#72245725, C2×#72245709, … |
| maximal normal subgroup of | C2×#72241395, C2×#72245725, C2×#72245709, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(7,17,10)(8,15,13)(11,20,19)(14,22,18), (4,6)(9,16)(10,19)(12,21)(13,18)(15,22)(17,20), (3,4)(5,6)(7,8,11,14)(9,15,16,22)(10,18,19,13)(12,20,21,17));