Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#2196 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C23×C32 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C22×C32 ◁ C2×C32×D4 ◁ G | |
1062 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2,3)(7,8)(9,11)(10,12)(13,14);(4,5,6)(8,11)(12,14);(9,13)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3256);