Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32×C4 · Schur multiplier (H2) = C26×C3 |
| direct product | C32×#2879 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C24 |
| commutator | C22 | C23×C32×C4 |
| Frattini, cosocle | C2×C4 | C24×C32 |
| socle | C22×C32 | C23×C4 |
| prime radical | C22×C32×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ G | |
2292 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)(11,12,14,18,15,17,21,24)(13,16,20,23,19,22,25,26);(4,5,6)(8,10)(11,13)(12,16)(14,20)(15,19)(17,22)(18,23)(21,25)(24,26);(13,19)(16,22)(20,25)(23,26);(12,17)(16,22)(18,24)(23,26)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3939);