Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C8 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#1336 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | #102 |
| commutator | C8 | C2×C32×C8 |
| Frattini, cosocle | C4×C8 | C22×C32 |
| socle | C2×C32 | D4⋊C8 |
| prime radical | C32×#135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C2×C32×C8 ◁ C32×C4×C8 ◁ G | |
288 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 132 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 — 2 generators on 22 points:
(1,2,3)(7,8,11,15,12,16,22,21,13,17,10,19,20,9,18,14);(4,5,6)(7,9)(8,12)(10,14)(11,21)(13,16)(15,18)(17,20)(19,22)288 conjugacy classes — view the full character table.
| Fitting subgroup of | C3⋊D3×#1336, #72255423 |
| maximal subgroup of | C32×#12068, C3⋊D3×#1336, #72255423 |
| maximal normal subgroup of | C32×#12068, C3⋊D3×#1336, #72255423 |
| maximal quotient of | C32×#7070 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2396);