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C7×C103 #10624372

order721
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C22×C32×C17 (order 612) · Inn(G) = C1 · Out(G) = C22×C32×C17
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC721
abelianC7×C103

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 110
transitive degreeμt(G) = 721

A minimal faithful permutation representation — 1 generator on 110 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)

Conjugacy classes & character table

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

Relationships

center ofD3×C7×C103, C7×AGL(1,𝔽5)×C103, C7×ΓL(1,𝔽8)×C103, …
derived subgroup ofD721, #14524189, #14524190, …
Fitting subgroup ofD7×C103, C7×D103, D721, …
socle ofD7×C103, C7×D103, D721, …
maximal subgroup ofD7×C103, C7×D103, D721, …
maximal normal subgroup ofD7×C103, C7×D103, D721, …
maximal quotient ofC2×C7×C103, C3×C7×C103, C5×C7×C103, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(721, 1);