Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×Q8 ⋊ C2×C3 |
| Hall | #1161 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #41 |
| Sylow | C7 ⋊ C3×#41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C7 | C23×C3 |
| Fitting | C7×Q8 | C2×C3 |
| hypercenter | Q8 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#41 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,7)(3,4,6)(8,10,11,14)(9,12,13,15);(8,9,11,13)(10,15,14,12);(2,3)(4,6)(5,7)(10,14)(12,15)Polycyclic presentation:
2^3,3,2,7|b^2=e,c^2=e,c^a=ce,f^a=f^6,c^b=ce,f^d=f^435 conjugacy classes — view the full character table.
| maximal subgroup of | #6477⋊C3, #6479⋊C3, #6481⋊C3, … |
| maximal normal subgroup of | #6477⋊C3, #6479⋊C3, #6481⋊C3, … |
| maximal quotient of | #6453⋊C3, #6455⋊C3, #6456⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 128);