Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#3054 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| socle | C22×C32 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C23×C32×C4 ◁ G | |
1500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2,3)(7,8)(9,10)(11,12)(13,20)(14,18)(15,24)(16,19)(17,23)(21,26)(22,25);(4,5,6)(11,13,15,23,16,21,14,22)(12,17,18,20,19,25,24,26);(8,10)(13,21)(17,25)(20,26)(22,23);(11,14,16,15)(12,18,19,24)(13,22,21,23)(17,20,25,26)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4114);