Finite Group Databasebeta

← Prev (#67059) Next (#67061) →

Cayley diagram

C5×C71 #67060

order355
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C4×C5×C7 (order 280) · Inn(G) = C1 · Out(G) = C2×C4×C5×C7
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC355
abelianC5×C71

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 76
transitive degreeμt(G) = 355

A minimal faithful permutation representation — 1 generator on 76 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)

Conjugacy classes & character table

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

Relationships

center ofC5×D3×C71, C5×D5×C71, C5×A4×C71, …
derived subgroup ofD355, Dic355, C71⋊AGL(1,𝔽5), …
Fitting subgroup ofD5×C71, C5×D71, D355, …
socle ofD5×C71, C5×D71, D355, …
maximal subgroup ofD5×C71, C5×D71, D355, …
maximal normal subgroup ofD5×C71, C5×D71, D355, …
maximal quotient ofC2×C5×C71, C3×C5×C71, C25×C71, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(355, 2);