Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C35 |
| direct product | C2×C3×#92258 |
| Fitting | C2×C3×C32⋊H(3,𝔽3) ⋊ C2 |
| derived | C33 ⋊ C22×C33 |
| Sylow | C3×C32⋊H(3,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #92258 |
| commutator, cosocle, hypocenter | C33 | C22×C33 |
| Frattini | C32 | C2×C33×D3 |
| Fitting | C2×C3×C32⋊H(3,𝔽3) | C2 |
| socle | C2×C33 | C32×D3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C3 | |
2800 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 2800 conjugacy classes of subgroups — too many to draw. Only its 248 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 23 points:
(5,10,12,7,8,11)(6,9)(14,16)(15,18)(17,19)(20,21);(4,5,7)(6,8,11)(9,12,10)(13,14,20)(15,17,19)(16,18,21)(22,23);(1,2,3)234 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#13091503 |
| maximal quotient of | C2×C3×#13091108, C2×#72221918, C2×#72221895, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,10,12,7,8,11)(6,9)(14,16)(15,18)(17,19)(20,21), (4,5,7)(6,8,11)(9,12,10)(13,14,20)(15,17,19)(16,18,21)(22,23), (1,2,3));