Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C23×C3×C4 |
| direct product | C2×C32×#482 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| prime radical | C2×C32×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C23×C32×C4 ◁ G | |
1404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(19,20,21);(3,4)(5,8)(6,7)(9,10)(12,15)(13,16)(17,18)(22,23,24);(4,7)(8,10)(11,12,14,15)(13,16,17,18);(3,5,6,9)(4,8,7,10)(11,13,14,17)(12,16,15,18)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4147);