Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C32 · Schur multiplier (H2) = C214×C3 |
| direct product | C22×C32×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C24 |
| commutator, Frattini, cosocle | C2 | C26×C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ G | |
19680 conjugacy classes of subgroups; only the 17016 normal subgroups are shown.
This group has 19680 conjugacy classes of subgroups — too many to draw. Only its 17016 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) = 288 |
A minimal faithful permutation representation — 6 generators on 18 points:
(1,2)(13,14,15);(3,4)(16,17,18);(5,6)(7,11)(8,9)(10,12);(6,9)(10,11);(5,6)(7,10)(8,9)(11,12);(5,7)(6,10)(8,12)(9,11)612 conjugacy classes — view the full character table.
| maximal subgroup of | C32×C22⋊#997 |
| maximal normal subgroup of | C32×C22⋊#997 |
| maximal quotient of | C32×#62817 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4651);