Finite Group DatabasebetaFinite Group Databasebeta| order | 52 |
| isoclinism | isoclinic class of #261 |
| automorphism | Aut(G) = AGL(1,𝔽13) (order 156) · Inn(G) = G · Out(G) = C3 |
| holomorph | G ⋊ Aut(G) = #72214122 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C13 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C13 | C4 |
| prime radical, cosocle | D13 | C2 |
| LCS | G ▷ C13 | |
| UCS | C1 | |
6 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 13 |
A minimal faithful permutation representation — 2 generators on 13 points:
(2,3,4,6)(5,8,9,13)(7,11,12,10);(1,2,5,8,12,6,10,11,3,7,13,9,4)Polycyclic presentation:
2^2,13|a^2=b,c^a=c^8,c^b=c^12| C1 | C2 | C3 | C4 | C5 | C6 | C7 | |
|---|---|---|---|---|---|---|---|
| size | 1 | 4 | 4 | 4 | 13 | 13 | 13 |
| order | 1 | 13 | 13 | 13 | 2 | 4 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | 1 | -1 | ||
| χ4 | 1 | 1 | 1 | 1 | -1 | ||
| χ5 | 4 | −ζ6−ζ22−ζ30−ζ46 | −ζ14−ζ18−ζ34−ζ38 | −ζ2−ζ10−ζ42−ζ50 | 0 | 0 | 0 |
| χ6 | 4 | −ζ14−ζ18−ζ34−ζ38 | −ζ2−ζ10−ζ42−ζ50 | −ζ6−ζ22−ζ30−ζ46 | 0 | 0 | 0 |
| χ7 | 4 | −ζ2−ζ10−ζ42−ζ50 | −ζ6−ζ22−ζ30−ζ46 | −ζ14−ζ18−ζ34−ζ38 | 0 | 0 | 0 |
Here ζ is a primitive root of unity of order 52 (ζ52 = 1); i = ζ13.
| maximal subgroup of | C2×#261, AGL(1,𝔽13), C3×#261, … |
| maximal normal subgroup of | C2×#261, AGL(1,𝔽13), C3×#261, … |
| maximal quotient of | #1057, C2×#261, C3×#261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(52, 3);