Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #3788 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C32×C8 ⋊ C4 |
| Sylow | C32 ⋊ #97 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×#157 |
| commutator, socle | C2×C32 | C42 |
| Fitting | C32×C8 | C4 |
| prime radical | C2×C3⋊D3 | C2×C4 |
| cosocle | C4×C3⋊D3 | C22 |
| hypercenter | C8 | #157 |
| hypocenter | C32 | #97 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C8 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,3,6)(2,5,4)(7,8,9,12,10,11,13,14);(1,2)(3,5,6,4)(8,11)(12,14)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=d,d^2=e,b^a=be,f^a=fg^2,g^a=f^2g^2,f^c=f^2,g^c=g^236 conjugacy classes — view the full character table.
| maximal subgroup of | #10590136, #10593282, #10593291, … |
| maximal normal subgroup of | #10590136, #10593282, #10593291, … |
| maximal quotient of | #10590136, #10590168, #10590169, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 415);