Finite Group DatabasebetaFinite Group Databasebeta| order | 30720 |
| isoclinism | isoclinic class of #72224460 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C29 ⋊ A5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #72209715 |
| commutator, hypocenter | G | C1 |
| Fitting, cosocle | C29 | A5 |
| socle | C25 | #11757162 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
At least 82502 conjugacy classes of subgroups; only the 7 normal subgroups are shown.
This group has at least 82502 conjugacy classes of subgroups — too many to draw. Only its 7 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 20 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,9,12,4,15,7)(2,11,18)(3,19,10,8,14,6)(5,17,13);(1,3,13,4,8,18)(2,9,10)(5,15,6)(7,17,16)(11,20,12)(14,19)52 conjugacy classes — view the full character table.
| maximal quotient of | #72266678, #72266680, #72266682, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,9,12,4,15,7)(2,11,18)(3,19,10,8,14,6)(5,17,13), (1,3,13,4,8,18)(2,9,10)(5,15,6)(7,17,16)(11,20,12)(14,19));