Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#72985 |
| Fitting | C22×C3×#95 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C2×#857 |
| commutator, cosocle, hypocenter | #95⋊C3 | C23×C3 |
| Frattini | C23 | C22×C3×S4 |
| Fitting | C22×C3×#95 | D3 |
| socle | C24×C3 | C2×S4 |
| hypercenter | C23×C3 | #857 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 | |
1409 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 1409 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) = 29 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 29 points:
(3,12,26,7,5,24)(4,17,23)(6,15,19,16,14,13)(8,9,25)(10,22,20)(11,18,21)(27,28,29);(1,2);(3,4)(5,11,12,20)(6,10)(7,8)(9,19)(13,17)(14,25,15,23)(16,18)(21,24)(22,26)144 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#72209884 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,12,26,7,5,24)(4,17,23)(6,15,19,16,14,13)(8,9,25)(10,22,20)(11,18,21)(27,28,29), (1,2), (3,4)(5,11,12,20)(6,10)(7,8)(9,19)(13,17)(14,25,15,23)(16,18)(21,24)(22,26));