Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | D3×#3027⋊C3 |
| Fitting | C3×#3027 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D3×#120⋊C3 |
| commutator, cosocle, hypocenter | C3×H(5,𝔽2) | C23×C3 |
| Frattini | C23 | C22×D3×A4 |
| Fitting | C3×#3027 | C2×C3 |
| socle | C2×C3 | C22×#120⋊C3 |
| hypercenter | C4 | D3×#120⋊C3 |
| LCS | G ▷ C3×H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ C4 | |
2006 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 2006 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(2,3)(6,13,7)(10,18,11)(12,15,14)(16,19,17);(1,2,3)(4,15)(5,18)(6,14,12,7)(8,13)(9,19)(10,11,16,17);(4,5,8,9)(6,10,12,16)(7,11,14,17)(13,18,15,19)69 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#11714184, D3×#11714174, D3×#60399⋊C3 |
| maximal normal subgroup of | D3×#11714184, D3×#11714174, D3×#60399⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(6,13,7)(10,18,11)(12,15,14)(16,19,17), (1,2,3)(4,15)(5,18)(6,14,12,7)(8,13)(9,19)(10,11,16,17), (4,5,8,9)(6,10,12,16)(7,11,14,17)(13,18,15,19));