Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#11714171 |
| Fitting | C2×C3×B(4,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C2×#1020⋊C2 |
| commutator, cosocle, hypocenter | #997 | C23×C3 |
| Frattini | C23 | C22×C3×S4 |
| Fitting | C2×C3×B(4,𝔽2) | D3 |
| hypercenter | C22×C3 | #1020⋊C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 | |
1226 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 1226 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,14,7,6)(5,18,11,12)(8,15,17,13)(9,19,16,10);(6,17,14,13,7,15)(10,16,19,12,11,18);(4,5)(6,12)(7,16)(8,9)(10,13)(11,17)(14,18)(15,19)102 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#11714171, #72275082, C3×#72217027 |
| maximal normal subgroup of | D3×#11714171, #72275082, C3×#72217027 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,14,7,6)(5,18,11,12)(8,15,17,13)(9,19,16,10), (6,17,14,13,7,15)(10,16,19,12,11,18), (4,5)(6,12)(7,16)(8,9)(10,13)(11,17)(14,18)(15,19));