Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C3×#73027 |
| Fitting | C2×C3×B(4,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C3 | #1020⋊C2 |
| commutator, cosocle, hypocenter | #997 | C23×C3 |
| Frattini | C23 | C22×C3×S4 |
| Fitting | C2×C3×B(4,𝔽2) | D3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22×C3 | |
1308 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 1308 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(1,2)(3,4,5)(6,8,15)(7,12,19)(10,14,20)(11,17,21);(7,11)(8,14)(9,16)(15,20);(6,7)(8,13)(9,12)(10,11)(14,18)(15,19)(16,17)(20,21)102 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((1,2)(3,4,5)(6,8,15)(7,12,19)(10,14,20)(11,17,21), (7,11)(8,14)(9,16)(15,20), (6,7)(8,13)(9,12)(10,11)(14,18)(15,19)(16,17)(20,21));