Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×A4×#126⋊C3 |
| Fitting, Sylow | C23×#126 ⋊ C32 |
| derived | C22×C42 ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×#126⋊C3 |
| commutator, cosocle, hypocenter | C22×C42 | C22×C32 |
| Frattini | C22 | C22×A42 |
| Fitting | C23×#126 | C32 |
| socle | C25 | C2×C3×A4 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 ◁ C2 | |
952 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 952 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 72 |
A minimal faithful permutation representation — 2 generators on 22 points:
(3,6)(4,5)(7,15,8,12,20,17)(9,14,21)(10,22,18,11,16,13);(1,2)(4,6,5)(8,14,15)(9,20,18)(10,12,11)(13,21,22)(16,17,19)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72213872, C2×A4×#560⋊C3, #72258729, … |
| maximal normal subgroup of | C2×A4×#560⋊C3 |
| maximal quotient of | C2×#72213872, #72258729, #126⋊C32, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,6)(4,5)(7,15,8,12,20,17)(9,14,21)(10,22,18,11,16,13), (1,2)(4,6,5)(8,14,15)(9,20,18)(10,12,11)(13,21,22)(16,17,19));