Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#1594 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C2×D4 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C42 | C23×C32 |
| socle | C22×C32 | C4×D4 |
| prime radical | C2×C32×D8 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C32×C4×C8 ◁ G | |
528 conjugacy classes of subgroups; only the 252 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 252 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) = 576 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(7,8,10,12)(9,11,14,13)(15,16,19,17,18,20,22,21);(4,5,6)(8,11)(9,14)(12,13)(16,20)(17,21);(8,12)(11,13)(15,17,19,20,18,21,22,16)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2654);