Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×((C3⋊S4)⋊C3)⋊C2 (order 1728) · Inn(G) = C2×C3⋊S4 · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C24×C3 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊S4 |
| commutator, cosocle | C2×C3×A4 | C22 |
| Fitting | C24×C3 | D3 |
| socle | C23×C3 | C2×D3 |
| hypercenter | C22 | C3⋊S4 |
| hypocenter | C3×A4 | D4 |
| LCS | G ▷ C2×C3×A4 ▷ C3×A4 | |
| UCS | C1 ◁ C2 ◁ C22 | |
156 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 11 points:
(2,4)(5,6)(7,8)(9,10,11);(1,2)(3,4)(6,7)(10,11);(6,7,8)Polycyclic presentation:
2^3,3^2,2^2|b^a=bc,d^a=d^2,e^a=e^2,f^a=g,g^a=f,f^d=g,g^d=fg30 conjugacy classes — view the full character table.
| maximal subgroup of | #10596463, #10596467, D3×GL(2,ℤ/4), … |
| maximal normal subgroup of | #10596463, #10596467, D3×GL(2,ℤ/4), … |
| maximal quotient of | #10593645, #10593647, #10593648, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 917);