Finite Group DatabasebetaFinite Group Databasebeta| order | 3840 |
| isoclinism | isoclinic class of #11757163 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | C22×#11757163 |
| Fitting, socle | C26 ⋊ A5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #11757163 |
| commutator, hypocenter | #11757163 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C26 | A5 |
| cosocle | C24 | C22×A5 |
| LCS | G ▷ #11757163 | |
| UCS | C1 ◁ C22 | |
1369 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 1369 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 20 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2)(4,12,8)(5,9,6);(3,9,10,8,12,7,4,11,5,6)48 conjugacy classes — view the full character table.
| derived subgroup of | A4×#11757163 |
| maximal subgroup of | A4×#11757163, C2×#72218849, C22×#11757163⋊C2, … |
| maximal normal subgroup of | A4×#11757163, C22×#11757163⋊C2 |
| maximal quotient of | C2×D3×#11757163 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,12,8)(5,9,6), (3,9,10,8,12,7,4,11,5,6));