Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#85557 |
| Fitting | C2×C3×H(3,𝔽4) ⋊ D3 |
| derived | #5465 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #5934 |
| commutator, cosocle, hypocenter | #5465 | C22×C3 |
| Frattini | C22 | C2×C3×#1020 |
| Fitting | C2×C3×H(3,𝔽4) | D3 |
| hypercenter | C23×C3 | #1020 |
| LCS | G ▷ #5465 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 | |
1245 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 1245 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2)(3,4,5)(8,9,18)(11,20,14)(12,21,19)(15,16,17);(6,14,17,19)(7,18,20,8)(9,11,10,21)(12,16,13,15)96 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#11921467 |
| maximal quotient of | C2×C3×#72209884 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4,5)(8,9,18)(11,20,14)(12,21,19)(15,16,17), (6,14,17,19)(7,18,20,8)(9,11,10,21)(12,16,13,15));