Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C3×#85559 |
| Fitting | C2×C3×H(3,𝔽4) ⋊ D3 |
| derived | #5464 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #85559 |
| commutator, cosocle, hypocenter | #5464 | C22×C3 |
| Frattini | C22 | C2×C3×#1020 |
| Fitting | C2×C3×H(3,𝔽4) | D3 |
| socle | C23×C3 | #1020 |
| LCS | G ▷ #5464 | |
| UCS | C1 ◁ C2×C3 | |
1088 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2)(3,4,5)(7,17,19)(8,21,15)(9,10,18)(11,20,16)(12,13,14);(6,7,20,17,18,16,13,12)(8,10,15,19,11,9,14,21)72 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#11715753, C2×C3×#11921467 |
| maximal normal subgroup of | C2×C3×#11715753 |
| maximal quotient of | C2×C3×#11714279 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4,5)(7,17,19)(8,21,15)(9,10,18)(11,20,16)(12,13,14), (6,7,20,17,18,16,13,12)(8,10,15,19,11,9,14,21));