Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#3289 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C32×C4 ◁ G | |
1344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2,3)(7,8,13,19)(9,20,24,35)(10,16,26,34)(11,18,12,17)(14,27,31,23)(15,28,32,25)(21,29,38,30)(22,36,37,33);(4,5,6)(7,9,13,24)(8,14,19,31)(10,21,26,38)(11,27,12,23)(15,29,32,30)(16,33,34,36)(17,35,18,20)(22,28,37,25);(7,10)(8,15)(9,21)(11,25)(12,28)(13,26)(14,29)(16,17)(18,34)(19,32)(20,36)(22,27)(23,37)(24,38)(30,31)(33,35);(7,10)(8,16)(9,22)(11,28)(12,25)(13,26)(14,30)(15,17)(18,32)(19,34)(20,36)(21,27)(23,38)(24,37)(29,31)(33,35)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4349);