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Cayley diagram

C2×C53 #1072

order106
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C4×C13 (order 52) · Inn(G) = C1 · Out(G) = C4×C13
holomorphG ⋊ Aut(G) = C2×AGL(1,𝔽53)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC106
abelianC2×C53
multiplicative(ℤ/107)ˣ, (ℤ/214)ˣ

Subgroups

subgroupG/H
centerGC1
commutator, FrattiniC1G
LCSG ▷ C1
UCSC1 ◁ G

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 55
transitive degreeμt(G) = 106

A minimal faithful permutation representation — 1 generator on 55 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55)

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

center ofD4×C53, Q8×C53, Dic3×C53, …
derived subgroup ofDic106, D212, C22⋊D53, …
Frattini subgroup ofC3×C4×C2809, Dic3×C2809, C3×Dic2809, …
Fitting subgroup ofDic53, C2×D53, #88324, …
socle ofDic53, C4×C53, C2×D53, …
maximal subgroup ofDic53, C4×C53, C2×D53, …
maximal normal subgroup ofDic53, C4×C53, C2×D53, …
maximal quotient ofC4×C53, C22×C53, D3×C53, …
automorphism group ofC107, C2×C107
outer automorphism group ofC107, C2×C107

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(106, 2);