Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×((C22⋊C9)⋊C2)⋊C3 (order 864) · Inn(G) = C2×(C22⋊C9)⋊C2 · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×(C22⋊C9)⋊C2 |
| commutator, cosocle | C2×C22⋊C9 | C22 |
| Frattini | C2×C3 | C2×S4 |
| Fitting | C24×C3 | D3 |
| socle | C23×C3 | C2×D3 |
| hypercenter | C22 | (C22⋊C9)⋊C2 |
| hypocenter | C22⋊C9 | D4 |
| LCS | G ▷ C2×C22⋊C9 ▷ C22⋊C9 | |
| UCS | C1 ◁ C2 ◁ C22 | |
108 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,4)(5,6,8)(9,14,16,15,10,12,11,17,13);(1,2)(3,4)(6,7)(10,12)(11,15)(13,14)(16,17)Polycyclic presentation:
2^3,3^2,2^2|d^3=e^2,b^a=bc,d^a=d^2e,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 | #10593087, D4×(C22⋊C9)⋊C2, #10593089, … |
| maximal normal subgroup of | #10593087, D4×(C22⋊C9)⋊C2, #10593089, … |
| maximal quotient of | ((C22⋊C9)⋊C4)⋊C4, (C2×(C22⋊C9)⋊C2)⋊C4, #10589516, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 342);