Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C42 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C32×#1297 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C22⋊C4 |
| commutator | C2×C4 | C32×C42 |
| Frattini, cosocle | C23×C4 | C22×C32 |
| socle | C22×C32 | #95 |
| prime radical | C24×C32 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C23×C32×C4 ◁ G | |
738 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 738 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3)(7,8,10,13,11,14,9,12)(15,16)(17,21)(18,20)(19,22);(4,5,6)(8,12,14,13)(15,17)(16,19,20,21)(18,22)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2357);