Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C22×#2202⋊C3 |
| Fitting, Sylow | C22×#2202 ⋊ C3 |
| derived | H(3,𝔽4) ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2)⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽4) | C23×C3 |
| Frattini | C2×Q8 | C23×A4 |
| Fitting | C22×#2202 | C3 |
| hypercenter | C24 | #120⋊C3 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 ◁ C23 ◁ C24 | |
1958 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 1958 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(3,4)(7,15,18)(8,17,16)(11,20,12)(13,14,19);(1,2)(6,20,10,14)(7,16)(11,19,13,12)(17,18);(5,6,9,10)(7,11,16,13)(8,14,15,20)(12,17,19,18)76 conjugacy classes — view the full character table.
| maximal subgroup of | #72271440 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(7,15,18)(8,17,16)(11,20,12)(13,14,19), (1,2)(6,20,10,14)(7,16)(11,19,13,12)(17,18), (5,6,9,10)(7,11,16,13)(8,14,15,20)(12,17,19,18));