Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C2×#10596417 (order 1152) · Inn(G) = C2×GO+(4,𝔽2) · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×GO+(4,𝔽2) |
| commutator, cosocle | C3⋊Dic3 | C23 |
| Fitting | C22×C32 | D4 |
| socle | C2×C32 | C2×D4 |
| hypercenter | C22 | GO+(4,𝔽2) |
| hypocenter | C32 | #135 |
| LCS | G ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C22 | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,3,4)(2,5,6)(8,13)(9,16)(14,18)(17,19);(3,4)(7,9)(8,12)(10,19)(11,16)(13,21)(14,22)(15,18)(17,20);(1,2)(3,6)(4,5)(7,8)(9,15)(10,18)(11,13)(12,17)(14,20)(16,22)(19,21)Polycyclic presentation:
2^5,3^2|d^2=e,b^a=bd,c^a=ce,d^a=de,f^a=f^2,c^b=ce,d^b=de,f^b=g,g^b=f,f^d=f^2,g^d=g^224 conjugacy classes — view the full character table.
| maximal subgroup of | #10593167, #10596500, #10596502, … |
| maximal normal subgroup of | #10593167, #10596500, #10596502, … |
| maximal quotient of | #10593353, #10593364, #10593373, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 885);