Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4×C8 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C32×#346 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32×C8 | C22 |
| commutator | C2 | C2×C32×C4×C8 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| socle, prime radical | C23×C32 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32×C8 ◁ G | |
600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(23,24,25);(7,8,9,13,10,14,15,20,11,12,16,18,17,19,21,22)(26,27,28);(3,4,5,6)(8,12)(13,18)(14,19)(20,22)720 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3166);