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Cayley diagram

C7×C47 #66506

order329
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C22×C3×C23 (order 276) · Inn(G) = C1 · Out(G) = C22×C3×C23
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC329
abelianC7×C47

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 329

A minimal faithful permutation representation — 1 generator on 54 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)

Conjugacy classes & character table

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

Relationships

center ofD3×C7×C47, C7×D5×C47, C7×A4×C47, …
derived subgroup ofD329, Dic329, D7×D47, …
Fitting subgroup ofD7×C47, C7×D47, D329, …
socle ofD7×C47, C7×D47, D329, …
maximal subgroup ofD7×C47, C7×D47, D329, …
maximal normal subgroup ofD7×C47, C7×D47, D329, …
maximal quotient ofC2×C7×C47, C3×C7×C47, C5×C7×C47, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(329, 1);