Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#3099 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| prime radical | C32×#537 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C2×C32×#41 ◁ G | |
1194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2,3)(7,8)(9,12)(10,11)(13,14)(15,16,20,24)(17,21,26,30)(18,23,19,28)(22,27,29,25);(4,5,6)(8,11)(12,14)(15,17)(16,21)(18,27)(19,25)(20,26)(22,28)(23,29)(24,30);(7,9,10,13)(8,12,11,14)(15,17)(16,22)(18,25)(19,27)(20,26)(21,28)(23,30)(24,29);(15,18,20,19)(16,23,24,28)(17,25,26,27)(21,22,30,29)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4159);