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) = C23×C3×C4 |
| direct product | C2×C32×H(3,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C42 |
| commutator | C4 | C2×C32×C42 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| socle | C22×C32 | #95 |
| prime radical | C24×C32 | D4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ G | |
1212 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)(5,11,13,17)(6,9)(7,10)(8,12,15,18)(14,16)(22,23,24);(3,5)(4,8,10,15)(6,12)(7,13)(9,11,16,17)(14,18)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2790);