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Cayley diagram

C5×C67 #66531

order335
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C3×C4×C11 (order 264) · Inn(G) = C1 · Out(G) = C2×C3×C4×C11
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC335
abelianC5×C67

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 72
transitive degreeμt(G) = 335

A minimal faithful permutation representation — 1 generator on 72 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)

Conjugacy classes & character table

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

Relationships

center ofC5×D3×C67, C5×D5×C67, C5×A4×C67, …
derived subgroup ofD335, Dic335, C67⋊AGL(1,𝔽5), …
Fitting subgroup ofD5×C67, C5×D67, D335, …
socle ofD5×C67, C5×D67, D335, …
maximal subgroup ofD5×C67, C5×D67, D335, …
maximal normal subgroup ofD5×C67, C5×D67, D335, …
maximal quotient ofC2×C5×C67, C3×C5×C67, C25×C67, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(335, 1);