Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of #66662 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | #1161 ⋊ C3 |
| Sylow | C7 ⋊ #41⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×D7 |
| commutator, Fitting, cosocle, hypocenter | C7×Q8 | C2×C3 |
| socle | C2×C7 | C2×A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C2 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,5,8)(3,13,12,14,9,7,10,15,4,6,11,16)(18,19)(20,21)(22,23);(1,11,5,9)(2,15,8,13)(3,4,10,12)(6,7,14,16)(17,18,20,22,23,21,19)Polycyclic presentation:
2,3,2^2,7,2|a^2=f,c^2=f,d^2=f,e^a=e^6,c^b=cd,d^b=c,d^c=df35 conjugacy classes — view the full character table.
| maximal subgroup of | #10621469, #10621470, #10621471, … |
| maximal normal subgroup of | #10621469, #10621470, #10621471, … |
| maximal quotient of | SL(2,𝔽3)×Dic7, C2×#66662, #11758969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 131);