Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#87516 |
| Fitting | C22×C3×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#1020 |
| commutator, cosocle, hypocenter | #997 | C23×C3 |
| Frattini | C2 | C22×C3×#1020 |
| Fitting | C22×C3×H(5,𝔽2) | D3 |
| hypercenter | C23×C3 | #1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 | |
3299 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 3299 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:
(3,4,5)(7,11)(8,18,15)(9,21,16)(12,13,19,17,14,20);(1,2)(6,8,10,16)(7,12,11,17)(9,21,18,15)(13,20,14,19);(6,7)(8,14)(9,17)(10,11)(12,18)(13,16)(15,20)(19,21)138 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#11714170 |
| maximal normal subgroup of | C2×C3×#11714170 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,5)(7,11)(8,18,15)(9,21,16)(12,13,19,17,14,20), (1,2)(6,8,10,16)(7,12,11,17)(9,21,18,15)(13,20,14,19), (6,7)(8,14)(9,17)(10,11)(12,18)(13,16)(15,20)(19,21));