Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#87519 |
| Fitting | C2×C3×#585 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | #1020 |
| commutator, cosocle, hypocenter | #997 | C23×C3 |
| Frattini | C2 | C22×C3×#1020 |
| Fitting | C2×C3×#585 | D3 |
| socle | C22×C3 | C2×#1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C3×C4 | |
3437 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 3437 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 21 points:
(3,5,8,4)(6,10,18,9,15,17,14,12,13,11,7,16)(19,20,21);(1,2)(3,9,8,11)(4,14,5,6)(7,16,15,17)(10,18,12,13);(6,7)(9,10)(11,12)(14,15)156 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×#87519, C2×C3×#11715839 |
| maximal normal subgroup of | C2×D3×#87519, C2×C3×#11715839 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,5,8,4)(6,10,18,9,15,17,14,12,13,11,7,16)(19,20,21), (1,2)(3,9,8,11)(4,14,5,6)(7,16,15,17)(10,18,12,13), (6,7)(9,10)(11,12)(14,15));