Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#1860 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | #120 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| prime radical | C2×C32×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C22×C32×C4 ◁ G | |
858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3)(7,8,10,9)(11,12,17,22)(13,23,14,25)(15,21,16,20)(18,26,19,24);(4,5,6)(7,9,10,8)(11,13,17,14)(12,18,22,19)(15,26,16,24)(20,25,21,23);(11,14,17,13)(12,19,22,18)(15,26,16,24)(20,25,21,23)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2920);