Finite Group DatabasebetaFinite Group Databasebeta| order | 30720 |
| isoclinism | isoclinic class of #72227581 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C24×2−1+4 ⋊ A5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #72209721 |
| commutator, hypocenter | G | C1 |
| Fitting, cosocle | C24×2−1+4 | A5 |
| socle | C25 | #11757163 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
11170 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 11170 conjugacy classes of subgroups — too many to draw. Only its 8 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 42 points:
(2,4)(5,8)(6,7)(9,10)(11,16,23,17,12,20,31,13)(14,26,39,21,18,32,36,24)(15,35,40,41,19,37,38,42)(22,34,33,25,28,27,30,29);(1,2,4,9,8)(3,6,7,5,10)(13,16,14,15,27)(17,20,18,19,34)(21,33,40,28,35)(22,37,24,30,38)(23,32,25,36,42)(26,29,39,41,31)52 conjugacy classes — view the full character table.
| maximal quotient of | #72267998, #72271339, #72271469, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,4)(5,8)(6,7)(9,10)(11,16,23,17,12,20,31,13)(14,26,39,21,18,32,36,24)(15,35,40,41,19,37,38,42)(22,34,33,25,28,27,30,29), (1,2,4,9,8)(3,6,7,5,10)(13,16,14,15,27)(17,20,18,19,34)(21,33,40,28,35)(22,37,24,30,38)(23,32,25,36,42)(26,29,39,41,31));