Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4⋊S5 |
| commutator, prime radical, cosocle, hypocenter | SL(2,𝔽3)×A5 | C2 |
| Fitting | Q8 | ΓL(2,𝔽4) |
| socle | C2×A5 | S4 |
| LCS | G ▷ SL(2,𝔽3)×A5 | |
| UCS | C1 ◁ C2 | |
177 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 177 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,4,2,5)(6,20,8,18,7,21,12,17)(9,15,16,11,10,19,14,13);(2,3,4)(6,10,14)(7,9,16)(15,17,20)(18,21,19)31 conjugacy classes — view the full character table.
| maximal quotient of | #72276835 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,2,5)(6,20,8,18,7,21,12,17)(9,15,16,11,10,19,14,13), (2,3,4)(6,10,14)(7,9,16)(15,17,20)(18,21,19));