Finite Group Databasebeta

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Cayley diagram

C7×C59 #88022

order413
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C22×C3×C29 (order 348) · Inn(G) = C1 · Out(G) = C22×C3×C29
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC413
abelianC7×C59

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 66
transitive degreeμt(G) = 413

A minimal faithful permutation representation — 1 generator on 66 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)

Conjugacy classes & character table

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

Relationships

center ofD3×C7×C59, C7×D5×C59, C7×D17×C59, …
derived subgroup ofD413, Dic413, D7×D59, …
Fitting subgroup ofD7×C59, C7×D59, D413, …
socle ofD7×C59, C7×D59, D413, …
maximal subgroup ofD7×C59, C7×D59, D413, …
maximal normal subgroup ofD7×C59, C7×D59, D413, …
maximal quotient ofC2×C7×C59, C3×C7×C59, C5×C7×C59, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(413, 1);