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Cayley diagram

C3×C107 #66303

order321
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C22×C53 (order 212) · Inn(G) = C1 · Out(G) = C22×C53
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC321
abelianC3×C107

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) = 321

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

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

Relationships

center ofC3×D3×C107, H(3,𝔽3)×C107, 3−1+2×C107, …
derived subgroup ofD321, Dic321, D3×D107, …
Fitting subgroup ofD3×C107, C3×D107, D321, …
socle ofD3×C107, C3×D107, D321, …
maximal subgroup ofD3×C107, C3×D107, D321, …
maximal normal subgroup ofD3×C107, C3×D107, D321, …
maximal quotient ofC2×C3×C107, C9×C107, C32×C107, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(321, 1);