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Cayley diagram

C2×C131 #63131

order262
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C5×C13 (order 130) · Inn(G) = C1 · Out(G) = C2×C5×C13
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC262
abelianC2×C131
multiplicative(ℤ/263)ˣ, (ℤ/526)ˣ

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 133
transitive degreeμt(G) = 262

A minimal faithful permutation representation — 1 generator on 133 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133)

Conjugacy classes & character table

262 conjugacy classes. The character table is too large to display.

Relationships

center ofD4×C131, Q8×C131, Dic3×C131, …
derived subgroup ofC22⋊D131, D524, Dic262, …
Fitting subgroup ofDic131, C2×D131, C2×C131⋊C5, …
socle ofDic131, C4×C131, C2×D131, …
maximal subgroup ofDic131, C4×C131, C2×D131, …
maximal normal subgroup ofDic131, C4×C131, C2×D131, …
maximal quotient ofC4×C131, C22×C131, D3×C131, …
automorphism group ofC263, C2×C263
outer automorphism group ofC263, C2×C263

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(262, 2);