Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C3×#11710729 |
| Fitting | C3×#62817 ⋊ C3 |
| derived | C26 ⋊ C22×C32 |
| Sylow | #62817 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #11710729 |
| commutator, cosocle, hypocenter | C26 | C22×C32 |
| Frattini | C22 | C22×C3×C22⋊A4 |
| Fitting | C3×#62817 | C3 |
| socle | C22×C3 | C22×C22⋊A4 |
| LCS | G ▷ C26 | |
| UCS | C1 ◁ C3 | |
3474 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 3474 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(4,6)(5,9)(7,18,8,13,14,12)(10,16,17,15,11,19);(1,2,3)(4,5)(6,19)(7,11)(8,10)(9,18)(12,16)(13,15)(14,17);(5,17)(9,19)(10,11)(15,16)108 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#72215828, C2×C3×#11710729, C3×#72215837, … |
| maximal normal subgroup of | C3×#72215828, C2×C3×#11710729, C3×#72215837, … |
| maximal quotient of | C2×C3×#11710729 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,6)(5,9)(7,18,8,13,14,12)(10,16,17,15,11,19), (1,2,3)(4,5)(6,19)(7,11)(8,10)(9,18)(12,16)(13,15)(14,17), (5,17)(9,19)(10,11)(15,16));