Finite Group DatabasebetaFinite Group Databasebeta| order | 11664 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C32 |
| direct product | C22×#72214331 |
| Fitting | C22×H(3,𝔽3)2 ⋊ C22 |
| derived, Sylow | H(3,𝔽3)2 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #72214331 |
| commutator, cosocle, hypocenter | H(3,𝔽3)2 | C24 |
| Frattini | C32 | C22×C3⋊D32 |
| Fitting | C22×H(3,𝔽3)2 | C22 |
| socle | C22×C32 | C3⋊D32 |
| LCS | G ▷ H(3,𝔽3)2 | |
| UCS | C1 ◁ C22 | |
13161 conjugacy classes of subgroups; only the 407 normal subgroups are shown.
This group has 13161 conjugacy classes of subgroups — too many to draw. Only its 407 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 324 |
A minimal faithful permutation representation — 4 generators on 22 points:
(3,4)(6,9,10)(8,12,13)(14,16,20)(15,17,22)(18,21,19);(5,6,8)(7,9,12,11,10,13)(14,19,18,20,17,15)(16,21);(6,8)(9,13)(10,12)(15,16)(19,22)(20,21);(1,2)208 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(6,9,10)(8,12,13)(14,16,20)(15,17,22)(18,21,19), (5,6,8)(7,9,12,11,10,13)(14,19,18,20,17,15)(16,21), (6,8)(9,13)(10,12)(15,16)(19,22)(20,21), (1,2));