Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C27 |
| direct product | C22×#3532⋊C3 |
| Fitting, Sylow | C22×#3532 ⋊ C3 |
| derived | H(3,𝔽4) ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #3532⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽4) | C23×C3 |
| Frattini | C22 | C23×C22⋊A4 |
| Fitting | C22×#3532 | C3 |
| socle | C24 | C2×C22⋊A4 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 ◁ C22 | |
15868 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 15868 conjugacy classes of subgroups — too many to draw. Only its 133 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 — 3 generators on 20 points:
(3,4)(6,11,7)(8,9,16)(10,17,13)(12,20,15)(14,19,18);(1,2)(6,19)(7,15)(10,12)(11,17)(13,14)(18,20);(1,2)(3,4)(5,11,9,17)(6,15,12,13)(7,19,14,10)(8,18,16,20)72 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#11715753, C22×#63011⋊C3, C2×#72340506, … |
| maximal normal subgroup of | C22×#11715753, C22×#63011⋊C3, C2×#72340506, … |
| maximal quotient of | C2×#72237716, C2×#72340507 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(6,11,7)(8,9,16)(10,17,13)(12,20,15)(14,19,18), (1,2)(6,19)(7,15)(10,12)(11,17)(13,14)(18,20), (1,2)(3,4)(5,11,9,17)(6,15,12,13)(7,19,14,10)(8,18,16,20));