Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#2155 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | #106 |
| commutator | C8 | C22×C32×C4 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| socle | C2×C32 | C2×#106 |
| prime radical | C23×C32 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C22×C32×C4 ◁ C22×C32×C8 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 38 points:
(1,2,3)(7,8)(9,18,21,32)(10,15)(11,24)(12,27)(13,17)(14,20,29,34)(16,25)(19,31,22,36)(23,30)(26,37)(28,33,38,35);(4,5,6)(7,9,11,20,12,22,25,35,13,21,26,34,10,19,23,33)(8,14,16,18,15,28,24,31,17,29,30,32,27,38,37,36);(7,10,13,12)(8,15,17,27)(9,19,21,22)(11,23,26,25)(14,28,29,38)(16,24,30,37)(18,31,32,36)(20,33,34,35)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3215);