Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×D3×#393⋊C3 |
| Fitting | C2×C3×#393 ⋊ C2×C3 |
| derived | C22×C3×Q8 ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×#120⋊C3 |
| commutator, cosocle, hypocenter | C22×C3×Q8 | C23×C3 |
| Frattini | C23 | C22×D3×A4 |
| Fitting | C2×C3×#393 | C2×C3 |
| socle | C24×C3 | C22×A4 |
| LCS | G ▷ C22×C3×Q8 | |
| UCS | C1 ◁ C22 | |
2947 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 2947 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 144 |
A minimal faithful permutation representation — 3 generators on 21 points:
(4,5)(8,9,13)(10,12,11)(15,18,16)(19,20,21);(3,4,5)(6,12,9,7)(8,11,13,10)(14,20,17,18)(15,16,19,21);(1,2)(6,7)(8,10)(9,12)(11,13)108 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×#35346⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5)(8,9,13)(10,12,11)(15,18,16)(19,20,21), (3,4,5)(6,12,9,7)(8,11,13,10)(14,20,17,18)(15,16,19,21), (1,2)(6,7)(8,10)(9,12)(11,13));