Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#87515 |
| Fitting | C22×C3×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#1020 |
| commutator, cosocle, hypocenter | #997 | C23×C3 |
| Frattini | C2 | C22×C3×#1020 |
| Fitting | C22×C3×H(5,𝔽2) | D3 |
| hypercenter | C23×C3 | #1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 | |
3521 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 3521 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 96 |
A minimal faithful permutation representation — 3 generators on 21 points:
(3,4,5)(7,11)(8,16,14)(9,20,15)(12,13,19,17,18,21);(1,2)(6,8)(7,12)(9,14)(10,15)(11,17)(13,19)(16,20)(18,21);(6,7)(8,13)(9,12)(10,11)(14,19)(15,18)(16,17)(20,21)138 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#11715839 |
| maximal normal subgroup of | C2×C3×#11715839 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,5)(7,11)(8,16,14)(9,20,15)(12,13,19,17,18,21), (1,2)(6,8)(7,12)(9,14)(10,15)(11,17)(13,19)(16,20)(18,21), (6,7)(8,13)(9,12)(10,11)(14,19)(15,18)(16,17)(20,21));