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Cayley diagram

C2×C127 #7011

order254
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C7×C9 (order 126) · Inn(G) = C1 · Out(G) = C2×C7×C9
holomorphG ⋊ Aut(G) = C2×AGL(1,𝔽127)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC254
abelianC2×C127

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 129
transitive degreeμt(G) = 254

A minimal faithful permutation representation — 1 generator on 129 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129)

Conjugacy classes & character table

254 conjugacy classes — view the full character table.

Relationships

center ofD4×C127, Q8×C127, Dic3×C127, …
derived subgroup ofC22⋊D127, D508, Dic254, …
Frattini subgroup ofDic16129
Fitting subgroup ofDic127, C2×D127, C2×C127⋊C3, …
socle ofDic127, C4×C127, C2×D127, …
maximal subgroup ofDic127, C4×C127, C2×D127, …
maximal normal subgroup ofDic127, C4×C127, C2×D127, …
maximal quotient ofC4×C127, C22×C127, D3×C127, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(254, 2);