Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C28×C3 |
| direct product | C2×C32×#392 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C32 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C32 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C32 ◁ G | |
5436 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);(8,10)(11,12)(13,14)(18,19,20);(4,6)(8,10)(12,14);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)396 conjugacy classes — view the full character table.
| Fitting subgroup of | C3×#10767815, #72252335, #72252358 |
| maximal subgroup of | C32×#15258, #72215014, C2×C3⋊D3×#392, … |
| maximal normal subgroup of | C32×#15258, #72215014, C2×C3⋊D3×#392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3444);