Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C24×C3×C4 |
| direct product | C2×C32×#469 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C23×C32×C4 ◁ G | |
1788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(15,16,17);(3,4)(5,6)(8,11)(9,12)(13,14)(18,19,20);(7,8,12,14,10,13,9,11);(4,6)(8,13)(11,14)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4112);