Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| isoclinism | isoclinic class of #560⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C211 |
| direct product | C23×#560⋊C3 |
| Fitting, Sylow | C23×#560 ⋊ C3 |
| derived | C42 ⋊ C25×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #560⋊C3 |
| commutator, cosocle, hypocenter | C42 | C25×C3 |
| Frattini | C22 | C25×A4 |
| Fitting | C23×#560 | C3 |
| socle | C25 | C22×A4 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C23 | |
17508 conjugacy classes of subgroups; only the 1138 normal subgroups are shown.
This group has 17508 conjugacy classes of subgroups — too many to draw. Only its 1138 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) = 96 |
A minimal faithful permutation representation — 5 generators on 22 points:
(5,6)(8,12,11)(9,16,19)(10,17,21)(13,20,14)(15,22,18);(3,4)(7,22,12,21)(8,20,11,16)(9,19,15,10)(13,18,17,14);(1,2)(3,4)(5,6)(9,13)(10,18)(14,19)(15,17)(16,20)(21,22);(9,15)(10,14)(13,17)(16,22)(18,19)(20,21);(1,2)136 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A4×#560⋊C3, C2×#72213963, #72258930 |
| maximal normal subgroup of | C2×A4×#560⋊C3, C2×#72213963, #72258930 |
| maximal quotient of | C2×#72214043, #72258937 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(8,12,11)(9,16,19)(10,17,21)(13,20,14)(15,22,18), (3,4)(7,22,12,21)(8,20,11,16)(9,19,15,10)(13,18,17,14), (1,2)(3,4)(5,6)(9,13)(10,18)(14,19)(15,17)(16,20)(21,22), (9,15)(10,14)(13,17)(16,22)(18,19)(20,21), (1,2));