Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C9 · Schur multiplier (H2) = C2×C4 |
| direct product | C9×#1275 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C42 |
| commutator | C4 | C4×C8×C9 |
| Frattini, cosocle | C2×C3×C42 | C22×C3 |
| socle | C22×C3 | C3×#95 |
| prime radical | C23×C3 | C3×C4⋊C4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ G | |
228 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,14,13,15,16,17)(18,19)(20,27,29,33)(21,25)(22,31)(23,26)(24,28,32,30);(18,20,22,30,23,29,21,28)(19,24,25,27,26,32,31,33)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 7);