Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Sylow | C32 ⋊ C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×C3⋊D3 |
| commutator, socle, prime radical, cosocle | C22×C32 | C23 |
| Fitting | C32×C42 | C2 |
| hypercenter | C42 | C3⋊D3 |
| hypocenter | C32 | C4⋊Q8 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C42 | |
204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 18 points:
(4,5,6)(7,8,9,10);(1,2,3)(11,13,14,18)(12,15,16,17);(2,3)(5,6)(8,10)(11,12,14,16)(13,17,18,15)Polycyclic presentation:
2^5,3^2|a^2=e,b^2=e,c^2=d,b^a=be,c^a=cd,f^a=f^2,g^a=g^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10589094, #10589096, #10589347, … |
| maximal normal subgroup of | #10589094, #10589096, #10589347, … |
| maximal quotient of | #10592355, #10592360, #10592379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 726);