Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of #4245 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C3×C7 ⋊ C22⋊C4 |
| Sylow | C7 ⋊ (C2×D3)⋊C4 |
| Sylow | C3 ⋊ (C2×D7)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D3×D7 |
| commutator | C2×C3×C7 | C2×C4 |
| Fitting, socle, cosocle | C22×C3×C7 | C22 |
| prime radical | C22×D21 | C2 |
| hypocenter | C3×C7 | C22⋊C4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,4,6,7,5,3)(9,10)(11,13,14,12)(16,18);(2,3)(4,5)(6,7)(8,9,10)(11,12,14,13)(15,16,17,18)Polycyclic presentation:
2^4,3,7|a^2=c,b^2=cd,b^a=bd,f^a=f^6,e^b=e^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10620888, #10620899, #10620907, … |
| maximal normal subgroup of | #10620888, #10620899, #10620907, … |
| maximal quotient of | #10620492, #10620495, #10620498, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 44);