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) = C22×C3×C4 |
| direct product | C32×#1812 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | #117 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| socle | C2×C32 | #380 |
| prime radical | C2×C32×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C2×C32×M16 ◁ G | |
780 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)(10,21,23,14)(11,24)(12,16)(13,19)(15,25)(18,31,33,29)(20,30)(22,28,37,36)(26,32,35,38)(27,34);(4,5,6)(7,9,11,19,12,20,25,34)(8,13,15,17,16,27,24,30)(10,18,22,32,23,33,37,38)(14,26,28,29,21,35,36,31);(7,10)(8,14)(9,18)(11,22)(12,23)(13,26)(15,28)(16,21)(17,29)(19,32)(20,33)(24,36)(25,37)(27,35)(30,31)(34,38)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2872);