Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C42 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C32×#347 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C42 |
| commutator | C4 | C2×C32×C42 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| socle | C22×C32 | #97 |
| prime radical | C23×C32 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ G | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(19,20,21);(3,4,5,9,6,11,13,17,7,10,12,18,14,8,15,16)(22,23,24);(4,8,10,11)(5,12)(9,17,18,16)(13,15)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3169);