Finite Group DatabasebetaFinite Group Databasebeta| order | 4320 |
| isoclinism | isoclinic class of #11761914 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | C22×#11761914 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, socle, hypercenter | C22×C3 | A6 |
| commutator, hypocenter | #11761914 | C22 |
| Frattini, cosocle | C3 | C22×A6 |
| LCS | G ▷ #11761914 | |
| UCS | C1 ◁ C22×C3 | |
382 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 15 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) = 72 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2)(5,6,9,8,7,11)(10,13,12)(14,16,15)(17,20,19,22,21,18);(3,4)(5,6,18,14,9,8,20,15,7,11,22,16)(10,21,13,19,12,17)68 conjugacy classes — view the full character table.
| derived subgroup of | A4×#11761914 |
| maximal subgroup of | A4×#11761914 |
| maximal normal subgroup of | A4×#11761914 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(5,6,9,8,7,11)(10,13,12)(14,16,15)(17,20,19,22,21,18), (3,4)(5,6,18,14,9,8,20,15,7,11,22,16)(10,21,13,19,12,17));