Finite Group DatabasebetaFinite Group Databasebeta| order | 3840 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C24×C4 ⋊ A5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #11757162 |
| commutator, prime radical, hypocenter | #14515300 | C2 |
| Frattini, socle | C2 | C2×#11757162 |
| Fitting | C24×C4 | A5 |
| cosocle | C25 | C2×A5 |
| LCS | G ▷ #14515300 | |
| UCS | C1 ◁ C4 | |
467 conjugacy classes of subgroups; only the 7 normal subgroups are shown.
This group has 467 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) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,8,10,12,2,11,4,16,3,5,7,19)(6,18,20,22,9,21,14,23,13,15,17,24);(1,7,13,17,10,4,2,20,9,3)(5,16,21,23,19,12,8,24,18,11)(6,14)(15,22)32 conjugacy classes — view the full character table.
| maximal subgroup of | #72265798, #72266252, #72267347 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,8,10,12,2,11,4,16,3,5,7,19)(6,18,20,22,9,21,14,23,13,15,17,24), (1,7,13,17,10,4,2,20,9,3)(5,16,21,23,19,12,8,24,18,11)(6,14)(15,22));