Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of #66665 |
| automorphism | Aut(G) = C2×A4×AGL(1,𝔽7) (order 1008) · Inn(G) = D7⋊A4 · Out(G) = C2×C3 |
| 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 | D7⋊A4 |
| 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)(3,7,11,14,4,13,10,16,9,6,12,15)(18,22,21,19,20,23);(1,9,5,11)(2,13,8,15)(3,12,10,4)(6,16,14,7)(17,18,21,22,20,23,19)Polycyclic presentation:
2,3,2^2,7,2|a^2=f,c^2=f,d^2=f,e^a=e^6,c^b=d,d^b=cd,e^b=e^4,d^c=df19 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#66665, #10621538, #10621539, … |
| maximal normal subgroup of | C2×#66665, #10621538, #10621539, … |
| maximal quotient of | Dic7⋊SL(2,𝔽3), C2×#66665, #11758972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 134);