Finite Group Databasebeta

← Prev (#3655) Next (#3657) →

Cayley diagram

C2×C71 #3656

order142
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C5×C7 (order 70) · Inn(G) = C1 · Out(G) = C2×C5×C7
holomorphG ⋊ Aut(G) = C2×AGL(1,𝔽71)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC142
abelianC2×C71

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 73
transitive degreeμt(G) = 142

A minimal faithful permutation representation — 1 generator on 73 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73)

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

center ofD4×C71, Q8×C71, Dic3×C71, …
derived subgroup ofDic142, D284, C22⋊D71, …
Frattini subgroup ofDic10082, D20164, Dic15123
Fitting subgroup ofDic71, C2×D71, C2×C71⋊C5, …
socle ofDic71, C4×C71, C2×D71, …
maximal subgroup ofDic71, C4×C71, C2×D71, …
maximal normal subgroup ofDic71, C4×C71, C2×D71, …
maximal quotient ofC4×C71, C22×C71, D3×C71, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(142, 2);