Finite Group DatabasebetaFinite Group Databasebeta| order | 196 |
| isoclinism | isoclinic class of #5997 |
| automorphism | Aut(G) = AΓL(1,𝔽49) (order 4704) · Inn(G) = G · Out(G) = C3×D4 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C7 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C72 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C72 | C4 |
| prime radical, cosocle | C7⋊D7 | C2 |
| LCS | G ▷ C72 | |
| UCS | C1 | |
14 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 14 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2)(3,5,6,10)(4,8,9,13)(7,11,12,14);(1,3,7,13,8,12,6)(2,9,5,14,11,10,4)Polycyclic presentation:
2^2,7^2|a^2=b,c^a=c^4d,d^a=c^4d^3,c^b=c^6,d^b=d^616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(196, 8);