Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C32×#1355 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | #329 |
| commutator | C42 | C2×C32×C4 |
| Frattini, cosocle | #97 | C22×C32 |
| prime radical | C32×#127 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C32×C4 ◁ C32×#97 ◁ G | |
264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3)(7,8,9,14,10,12,18,20,11,15,17,22,19,13,16,21);(4,5,6)(8,12)(9,16)(10,19)(13,15)(14,22)(17,18)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2415);