Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11711594 |
| Fitting | C2×#1790 ⋊ D3 |
| derived | #997 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | #1020⋊C2 |
| commutator, hypocenter | #997 | C22×C4 |
| Frattini | C24 | C22×S4 |
| Fitting | C2×#1790 | D3 |
| prime radical | C22×#5935 | C2 |
| cosocle | C2×#997 | C23 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C23 | |
1918 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 1918 conjugacy classes of subgroups — too many to draw. Only its 59 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)(7,19,11)(8,17,14)(9,18,16)(12,22,21);(3,5,6,4)(8,12)(10,15)(11,16)(18,19);(3,4,6,5)(7,8,9,12)(10,17,15,22)(11,14,16,21)(13,18,20,19)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72245603, C2×#72245595, C2×#72245678, … |
| maximal normal subgroup of | C2×#72245603, C2×#72245595, C2×#72245678, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(7,19,11)(8,17,14)(9,18,16)(12,22,21), (3,5,6,4)(8,12)(10,15)(11,16)(18,19), (3,4,6,5)(7,8,9,12)(10,17,15,22)(11,14,16,21)(13,18,20,19));