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Cayley diagram

C4×C81 #66310

order324
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C22×C27 (order 108) · Inn(G) = C1 · Out(G) = C22×C27
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC324
abelianC4×C81

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2×C27C2×C3
socle, prime radicalC2×C3C2×C27
LCSG ▷ C1
UCSC1 ◁ G

15 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 85
transitive degreeμt(G) = 324

A minimal faithful permutation representation — 1 generator on 85 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)

Conjugacy classes & character table

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

Relationships

center ofM16×C81, #41×C81, C3⋊C8×C81, …
derived subgroup ofDic324, #13046289, D648, …
Frattini subgroup ofC243⋊C8, C8×C243, D13608, …
Fitting subgroup ofC81⋊C8, Dic162, C4×D81, …
maximal subgroup ofC81⋊C8, C8×C81, Dic162, …
maximal normal subgroup ofC81⋊C8, C8×C81, Dic162, …
maximal quotient ofC8×C81, C2×C4×C81, C4×C243, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 2);