Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C32×D8 ⋊ C2 |
| Sylow | C32 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×C3⋊D3 |
| commutator, cosocle | C32×C4 | C23 |
| Frattini | C4 | C22×C3⋊D3 |
| Fitting | C32×D8 | C2 |
| socle | C2×C32 | C2×D4 |
| hypercenter | D8 | C3⋊D3 |
| hypocenter | C32 | #135 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(2,7)(3,10)(4,13)(6,14)(9,16)(11,12)(20,21,22);(1,3)(2,6)(4,10)(5,12)(7,9)(8,16)(11,13)(14,15)(17,18,19);(1,2)(3,9)(4,8)(5,7)(6,12)(10,14)(11,16)(13,15)(18,19)(21,22)Polycyclic presentation:
2^5,3^2|d^2=e,b^a=be,c^a=ce,f^a=f^2,g^a=g^2,c^b=cd,d^b=de,d^c=de42 conjugacy classes — view the full character table.
| maximal subgroup of | #10590192, #10590194, #10591026, … |
| maximal normal subgroup of | #10590192, #10590194, #10591026, … |
| maximal quotient of | #10592457, #10592465, #10592466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 769);