Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#3037 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| prime radical | C32×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C2×C32×C42 ◁ G | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2,3)(8,10)(11,12)(13,19)(14,22)(15,17)(16,20)(18,24)(21,26)(23,25);(4,5,6)(7,8,9,10);(11,13,15,21)(12,16,17,25)(14,23,24,20)(18,26,22,19);(12,17)(16,25)(18,22)(19,26)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4097);