Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C8 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#1331 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | D8 |
| commutator | C8 | C2×C32×C8 |
| Frattini, cosocle | C4×C8 | C22×C32 |
| socle | C22×C32 | #102 |
| prime radical | C2×C32×D8 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C32×C42 ◁ C32×C4×C8 ◁ G | |
330 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3)(7,8,9,11,10,12,13,14)(15,16)(17,20)(18,27)(19,23)(21,24)(22,25)(26,29)(28,30);(4,5,6)(16,20)(17,24)(18,25)(21,22)(23,29)(26,30)(27,28)396 conjugacy classes — view the full character table.
| Fitting subgroup of | C3⋊D3×#1331, #72254924 |
| maximal subgroup of | C32×#12045, C3⋊D3×#1331, #72254924 |
| maximal normal subgroup of | C32×#12045, C3⋊D3×#1331, #72254924 |
| maximal quotient of | C32×#7385 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2391);