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×#3288 |
| 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 | |
1392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3)(8,13)(10,11)(14,21)(15,19)(16,22)(17,20);(4,5,6)(7,8)(9,16)(10,19)(11,13)(12,15)(14,17)(18,21)(20,22);(7,8)(9,14)(10,13)(11,19)(12,15)(16,17)(18,22)(20,21);(7,9)(8,14)(10,20)(11,17)(12,18)(13,21)(15,22)(16,19)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4348);