Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#1416 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C32 | D16 |
| commutator | C16 | C2×C32×C4 |
| Frattini, cosocle | C2×C16 | C22×C32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C2×C32×C4 ◁ C2×C32×C8 ◁ C2×C32×C16 ◁ G | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 2 generators on 42 points:
(1,2,3)(7,8,9,10)(12,17)(13,21)(14,23)(15,27)(18,33)(19,36)(20,32)(22,26)(24,40)(25,30)(28,34)(29,31)(35,41)(37,38)(39,42);(4,5,6)(11,12,16,20)(13,22,25,39)(14,26,28,42)(15,29,27,19)(17,30,32,21)(18,34,35,23)(24,41,37,33)(31,38,36,40)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2476);