Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | A4×#5450 |
| Fitting, Sylow | C22×H(3,𝔽4) ⋊ C32 |
| derived | C26 ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C26 | C22×C32 |
| Frattini | C22 | C22×A42 |
| Fitting | C22×H(3,𝔽4) | C32 |
| socle | C24 | C22×C3×A4 |
| LCS | G ▷ C26 | |
| UCS | C1 | |
1021 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 1021 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,4,2)(5,20,16,7)(6,19,12,18)(8,15,9,14)(10,13,17,11);(2,4,3)(5,6)(7,10)(8,12,9,13,16,11)(14,17,15,18,20,19)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72258621, C2×A4×#5450 |
| maximal normal subgroup of | #72258621, C2×A4×#5450 |
| maximal quotient of | C2×A4×#5450 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,2)(5,20,16,7)(6,19,12,18)(8,15,9,14)(10,13,17,11), (2,4,3)(5,6)(7,10)(8,12,9,13,16,11)(14,17,15,18,20,19));