Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32×C4 · Schur multiplier (H2) = C24×C3×C4 |
| direct product | C2×C32×#387 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C32 | C23 |
| commutator | C22 | C23×C32×C4 |
| Frattini, cosocle | C23 | C24×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C32 ◁ G | |
2136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(19,20,21);(4,6)(7,8,10,9)(12,15)(13,17)(22,23,24);(7,9,10,8)(11,12)(13,16)(14,15)(17,18);(3,4)(5,6)(11,13,14,17)(12,16,15,18)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3348);