Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #3774 |
| automorphism | Aut(G) = C24×GO+(4,𝔽2) (order 1152) · Inn(G) = GO+(4,𝔽2) · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ #103 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | GO+(4,𝔽2) |
| commutator | C3⋊Dic3 | C2×C4 |
| Fitting, socle, prime radical | C22×C32 | D4 |
| cosocle | C2×C3⋊Dic3 | C22 |
| hypocenter | C32 | #103 |
| LCS | G ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,3)(2,5,6)(11,12,14,16)(13,18,17,15);(1,2)(3,6)(4,5)(7,8,9,10)(12,15)(13,17)(16,18)Polycyclic presentation:
2^5,3^2|a^2=d,b^2=e,c^2=e,b^a=bc,c^a=ce,f^a=f^2,c^b=ce,f^b=g,g^b=f,f^c=f^2,g^c=g^230 conjugacy classes — view the full character table.
| maximal subgroup of | #10593349, #10593352, #10593353, … |
| maximal normal subgroup of | #10593349, #10593352, #10593353, … |
| maximal quotient of | #10590032, #10590039, #10590040, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 388);