Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C32 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D32 |
| commutator, socle, cosocle | C2×C32 | C24 |
| Fitting | C32×Q8 | C22 |
| hypercenter | Q8 | D32 |
| hypocenter | C32 | 2−1+4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,18,19)(20,21,22);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15);(2,6)(7,13)(8,14)(15,16)(18,19);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(21,22)Polycyclic presentation:
2^5,3^2|c^2=e,d^2=e,b^a=be,g^a=g^2,f^b=f^2,d^c=de42 conjugacy classes — view the full character table.
| maximal subgroup of | #10593483, #10593484, #10594748, … |
| maximal normal subgroup of | #10593483, #10593484, #10594748, … |
| maximal quotient of | #10594316, #10594317, #10594327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 964);