Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×C32×#357 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | D8 |
| commutator | C8 | C22×C32×C4 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| prime radical | C22×C32×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C22×C32×C4 ◁ C22×C32×C8 ◁ G | |
816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(23,24,25);(3,4,5,6)(7,8)(9,12)(10,19)(11,15)(13,16)(14,17)(18,21)(20,22)(26,27,28);(8,12)(9,16)(10,17)(13,14)(15,21)(18,22)(19,20)396 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×C3⋊D3×#357, C2×#11797801 |
| maximal subgroup of | C2×C32×#2177, C2×C3⋊D3×#357, C2×#11797801 |
| maximal normal subgroup of | C2×C32×#2177, C2×C3⋊D3×#357, C2×#11797801 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3196);