Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = #11809017 (order 1152) · Inn(G) = C2×GO+(4,𝔽2) · Out(G) = D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22×C32 ⋊ D4 |
| Sylow | C32 ⋊ #120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×GO+(4,𝔽2) |
| commutator, cosocle | C2×C3⋊D3 | C23 |
| Fitting | C22×C32 | D4 |
| socle | C2×C32 | C2×D4 |
| hypercenter | C22 | GO+(4,𝔽2) |
| hypocenter | C32 | #120 |
| LCS | G ▷ C2×C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C22 | |
224 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 3 generators on 10 points:
(1,3,4)(2,5,6)(8,10);(3,4)(7,8)(9,10);(1,2)(3,6)(4,5)Polycyclic presentation:
2^5,3^2|b^a=bd,f^a=f^2,c^b=ce,f^b=g,g^b=f,f^d=f^2,g^d=g^227 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 889);