Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C3 |
No aliases for this group.
| Fitting | C32×#41 ⋊ C2 |
| Sylow | C32 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×C3⋊D3 |
| commutator, socle, cosocle | C2×C32 | C24 |
| Fitting | C32×#41 | C2 |
| hypercenter | #41 | C3⋊D3 |
| hypocenter | C32 | 2−1+4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ #41 | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,5,6,4)(2,8,9,13)(3,10,11,12)(7,15,16,14)(20,21,22);(1,4,6,5)(2,8,9,13)(3,10,11,12)(7,14,16,15)(17,18,19);(1,3,6,11)(2,7,9,16)(4,12,5,10)(8,15,13,14);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,16)(12,15)(18,19)(21,22)Polycyclic presentation:
2^5,3^2|b^2=e,c^2=e,d^2=e,d^a=de,f^a=f^2,g^a=g^2,c^b=ce57 conjugacy classes — view the full character table.
| maximal subgroup of | #10591668, #10591686, #10592525, … |
| maximal normal subgroup of | #10591668, #10591686, #10592525, … |
| maximal quotient of | #10595987, #10595989, #10595990, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 1015);