Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C8 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#1333 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | #102 |
| commutator | C8 | C2×C32×C8 |
| Frattini, cosocle | C4×C8 | C22×C32 |
| prime radical | C2×C32×D8 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C42 ◁ C32×C4×C8 ◁ G | |
324 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 144 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)(15,16,18,22,19,24,29,23)(17,25,27,21,28,20,26,30);(4,5,6)(16,20)(17,26)(18,29)(21,22)(23,30)(24,25)(27,28)288 conjugacy classes — view the full character table.
| maximal subgroup of | C32×#12067, C3⋊D3×#1333, #72222571, … |
| maximal normal subgroup of | C32×#12067, C3⋊D3×#1333, #72222571, … |
| maximal quotient of | C2×C32×#1333, C32×#7389, C32×#7390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2393);