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Cayley diagram

C7×C53 #67307

order371
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C3×C4×C13 (order 312) · Inn(G) = C1 · Out(G) = C2×C3×C4×C13
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC371
abelianC7×C53

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 60
transitive degreeμt(G) = 371

A minimal faithful permutation representation — 1 generator on 60 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)

Conjugacy classes & character table

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

Relationships

center ofD3×C7×C53, C7×D5×C53, C7×D19×C53, …
derived subgroup ofD371, Dic371, C7⋊#6157, …
Fitting subgroup ofD7×C53, C7×D53, D371, …
socle ofD7×C53, C7×D53, D371, …
maximal subgroup ofD7×C53, C7×D53, D371, …
maximal normal subgroup ofD7×C53, C7×D53, D371, …
maximal quotient ofC2×C7×C53, C3×C7×C53, C5×C7×C53, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(371, 1);