Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | D3×#73032 |
| Fitting | C3×#458 ⋊ C2×D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×#1020⋊C2 |
| commutator, cosocle, hypocenter | C3×#997 | C23 |
| Frattini | C23 | C2×D3×S4 |
| Fitting | C3×#458 | C2×D3 |
| socle | C2×C3 | C2×#1020⋊C2 |
| LCS | G ▷ C3×#997 | |
| UCS | C1 ◁ C2 | |
1292 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 1292 conjugacy classes of subgroups — too many to draw. Only its 42 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:
(1,3,10)(2,7,14)(5,11,16)(6,9,15)(18,19);(1,2,5,6)(3,9,11,7)(4,12,13,8)(10,15,16,14)(17,18,19);(2,6)(3,4)(7,12)(8,9)(11,13)(14,15)51 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#11714170, D3×#11714182, D3×#11714175 |
| maximal normal subgroup of | D3×#11714170, D3×#11714182, D3×#11714175 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,10)(2,7,14)(5,11,16)(6,9,15)(18,19), (1,2,5,6)(3,9,11,7)(4,12,13,8)(10,15,16,14)(17,18,19), (2,6)(3,4)(7,12)(8,9)(11,13)(14,15));