Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C26×C3 |
| direct product | C32×#1781 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C2×C22⋊C4 |
| commutator | C23 | C22×C32×C4 |
| Frattini, cosocle | C24 | C23×C32 |
| prime radical | C23×C32×D4 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C25×C32 ◁ G | |
2640 conjugacy classes of subgroups; only the 432 normal subgroups are shown.
This group has 2640 conjugacy classes of subgroups — too many to draw. Only its 432 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,3)(7,8)(9,10)(11,12)(13,17,18,15)(14,16);(4,5,6)(11,13)(12,15)(14,18)(16,17);(8,10)288 conjugacy classes — view the full character table.
| Fitting subgroup of | C3×#10710720, C3×#10710712, #72252320, … |
| maximal subgroup of | C32×#23900, #72214782, C3⋊D3×#1781, … |
| maximal normal subgroup of | C32×#23900, #72214782, C3⋊D3×#1781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2841);