Finite Group DatabasebetaFinite Group Databasebeta| order | 6480 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
| direct product | A5×#1088 |
| Fitting | H(3,𝔽3) ⋊ C4×A5 |
| derived | H(3,𝔽3)×A5 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | #157×A5 |
| commutator, hypocenter | H(3,𝔽3)×A5 | C4 |
| Fitting | H(3,𝔽3) | C4×A5 |
| socle | C3×A5 | #157 |
| prime radical | #272×A5 | C2 |
| cosocle | #272 | C2×A5 |
| LCS | G ▷ H(3,𝔽3)×A5 | |
| UCS | C1 ◁ C3 | |
232 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 90 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,8,5,18,2,16)(3,6,14)(4,7,15)(9,17,11,10,13,12)(19,20)(22,23);(1,7,18,13,5,4,16,9,2,15,8,11)(3,12,6,10,14,17)(20,21,22)70 conjugacy classes — view the full character table.
| maximal subgroup of | A5×#11761914 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,8,5,18,2,16)(3,6,14)(4,7,15)(9,17,11,10,13,12)(19,20)(22,23), (1,7,18,13,5,4,16,9,2,15,8,11)(3,12,6,10,14,17)(20,21,22));