Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#3183 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| prime radical | C32×#469 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C2×C32×C42 ◁ G | |
1110 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)(17,21)(18,25)(19,28)(20,23)(22,26)(24,30)(27,29);(4,5,6)(8,11)(12,14)(15,17,20,19)(16,21,23,28)(18,26,27,24)(22,29,30,25);(15,18,19,26,20,27,17,24)(16,22,21,25,23,30,28,29);(7,9,10,13)(8,12,11,14)(18,27)(22,30)(24,26)(25,29)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4243);