Finite Group Databasebeta

← Prev (#10620266) Next (#10620268) →

C5×C131 #10620267

order655
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C4×C5×C13 (order 520) · Inn(G) = C1 · Out(G) = C2×C4×C5×C13
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC655
abelianC5×C131

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 136
transitive degreeμt(G) = 655

A minimal faithful permutation representation — 1 generator on 136 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136)

Conjugacy classes & character table

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

Relationships

center ofC5×D3×C131, C5×ΓL(1,𝔽8)×C131, C5×D11×C131, …
derived subgroup ofD655, Dic655, C131⋊AGL(1,𝔽5), …
Fitting subgroup ofD5×C131, C5×D131, D655, …
socle ofD5×C131, C5×D131, D655, …
maximal subgroup ofD5×C131, C5×D131, D655, …
maximal normal subgroup ofD5×C131, C5×D131, D655, …
maximal quotient ofC2×C5×C131, C3×C5×C131, C25×C131, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(655, 2);