Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of #66560 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C3×C7 ⋊ Q16 |
| Sylow | C7 ⋊ #216 |
| Sylow | C3 ⋊ #1144 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4245 |
| commutator, Fitting, cosocle | C3×C4×C7 | C22 |
| Frattini, hypercenter | C4 | D3×D7 |
| socle, prime radical | C2×C3×C7 | D4 |
| hypocenter | C3×C7 | Q16 |
| LCS | G ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,4,6,7,5,3)(9,10)(11,13,15,21)(12,16,18,25)(14,23,24,20)(17,26,22,19);(2,3)(4,5)(6,7)(8,9,10)(11,12,15,18)(13,19,21,26)(14,22,24,17)(16,23,25,20)Polycyclic presentation:
2^4,3,7|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,f^a=f^6,c^b=cd,e^b=e^239 conjugacy classes — view the full character table.
| maximal subgroup of | #10620826, #10620838, #10620840, … |
| maximal normal subgroup of | #10620826, #10620838, #10620840, … |
| maximal quotient of | #10620505, #10620507, #10620524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 38);