Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C2×C3×C8 |
| direct product | C32×#1721 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C32 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C42 ◁ G | |
726 conjugacy classes of subgroups; only the 240 normal subgroups are shown.
This group has 726 conjugacy classes of subgroups — too many to draw. Only its 240 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(8,11)(9,13)(12,14)(16,19)(17,20)(21,22);(4,5,6)(7,8,9,11,10,12,13,14);(8,12)(11,14)(15,16,20,22,18,21,17,19)234 conjugacy classes — view the full character table.
| Fitting subgroup of | C3×D3×#1721, C3×#10784427 |
| maximal subgroup of | C32×#19745, #72214282, C3⋊D3×#1721, … |
| maximal normal subgroup of | C32×#19745, #72214282, C3⋊D3×#1721, … |
| maximal quotient of | 3−1+2×#1721 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2781);