Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#3046 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| prime radical | C2×C32×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×#117 ◁ G | |
1416 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)(11,12,16,20)(13,17,22,26)(14,19,15,24)(18,23,25,21);(4,5,6)(7,8,9,10)(11,13)(12,17)(14,23)(15,21)(16,22)(18,24)(19,25)(20,26);(11,13)(12,18)(14,21)(15,23)(16,22)(17,24)(19,26)(20,25);(11,14,16,15)(12,19,20,24)(13,21,22,23)(17,18,26,25)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4106);