Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#3057 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| prime radical | C32×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C2×C32×#41 ◁ G | |
1476 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)(12,16)(14,21)(17,22)(18,23)(20,25)(24,26);(4,5,6)(11,12,14,16,15,18,21,23)(13,17,20,22,19,24,25,26);(8,10)(11,13)(12,17)(14,20)(15,19)(16,22)(18,24)(21,25)(23,26);(13,19)(17,24)(20,25)(22,26)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4117);