Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#2238 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C23×C32 |
| prime radical | C32×#469 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C32×C4 ◁ C2×C32×M16 ◁ G | |
402 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 402 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) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(8,12)(9,15)(10,18)(13,20)(14,17)(16,22)(19,21);(4,5,6)(9,16)(12,20)(15,22)(17,21);(7,8,10,19,11,13,18,14)(9,17,15,20,16,21,22,12)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3298);