Finite Group DatabasebetaFinite Group Databasebeta| order | 68 |
| isoclinism | isoclinic class of #595 |
| automorphism | Aut(G) = AGL(1,𝔽17) (order 272) · Inn(G) = G · Out(G) = C4 |
| holomorph | G ⋊ Aut(G) = #72218905 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C17 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C17 | C4 |
| prime radical, cosocle | D17 | C2 |
| LCS | G ▷ C17 | |
| UCS | C1 | |
6 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 17 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,3,4,6)(5,8,9,13)(7,12,10,11)(14,16,17,15);(1,2,5,10,6,11,15,17,13,8,14,16,12,3,7,9,4)Polycyclic presentation:
2^2,17|a^2=b,c^a=c^13,c^b=c^16| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | |
|---|---|---|---|---|---|---|---|---|
| size | 1 | 4 | 4 | 4 | 4 | 17 | 17 | 17 |
| order | 1 | 17 | 17 | 17 | 17 | 2 | 4 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | 1 | 1 | -1 | ||
| χ4 | 1 | 1 | 1 | 1 | 1 | -1 | ||
| χ5 | 4 | −ζ18−ζ30−ζ38−ζ50 | −ζ6−ζ10−ζ58−ζ62 | −ζ14−ζ22−ζ46−ζ54 | −ζ2−ζ26−ζ42−ζ66 | 0 | 0 | 0 |
| χ6 | 4 | −ζ6−ζ10−ζ58−ζ62 | −ζ2−ζ26−ζ42−ζ66 | −ζ18−ζ30−ζ38−ζ50 | −ζ14−ζ22−ζ46−ζ54 | 0 | 0 | 0 |
| χ7 | 4 | −ζ14−ζ22−ζ46−ζ54 | −ζ18−ζ30−ζ38−ζ50 | −ζ2−ζ26−ζ42−ζ66 | −ζ6−ζ10−ζ58−ζ62 | 0 | 0 | 0 |
| χ8 | 4 | −ζ2−ζ26−ζ42−ζ66 | −ζ14−ζ22−ζ46−ζ54 | −ζ6−ζ10−ζ58−ζ62 | −ζ18−ζ30−ζ38−ζ50 | 0 | 0 | 0 |
Here ζ is a primitive root of unity of order 68 (ζ68 = 1); i = ζ17.
| maximal subgroup of | #3633, C2×#595, C3×#595, … |
| maximal normal subgroup of | #3633, C2×#595, C3×#595, … |
| maximal quotient of | #3624, C2×#595, C3×#595, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(68, 3);