Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C32×#1833 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | #117 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| socle | C2×C32 | #382 |
| prime radical | C23×C32 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C32×#117 ◁ G | |
516 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)(9,17,20,13)(10,21,23,36)(11,16)(12,15)(14,22,27,37)(18,31)(19,30,34,26)(24,28)(25,32)(29,38)(33,35);(4,5,6)(7,9,11,19,12,20,24,34)(8,13,16,26,15,17,28,30)(10,18,22,32,23,33,37,38)(14,25,21,35,27,29,36,31);(7,10)(8,14,15,27)(9,18)(11,22)(12,23)(13,25,17,29)(16,21,28,36)(19,32)(20,33)(24,37)(26,35,30,31)(34,38)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2893);