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Cayley diagram

C4×C59 #6630

order236
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C22×C29 (order 116) · Inn(G) = C1 · Out(G) = C22×C29
holomorphG ⋊ Aut(G) = D4×AGL(1,𝔽59)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC236
abelianC4×C59

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2C2×C59
socle, prime radicalC2×C59C2
LCSG ▷ C1
UCSC1 ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 63
transitive degreeμt(G) = 236

A minimal faithful permutation representation — 1 generator on 63 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)

Conjugacy classes & character table

236 conjugacy classes — view the full character table.

Relationships

center ofM16×C59, #41×C59, C3⋊C8×C59, …
derived subgroup of#11745660, D472, Dic236, …
Frattini subgroup ofDic13924, D27848
Fitting subgroup ofC59⋊C8, Dic118, C4×D59, …
maximal subgroup ofC59⋊C8, C8×C59, Dic118, …
maximal normal subgroup ofC59⋊C8, C8×C59, Dic118, …
maximal quotient ofC8×C59, C2×C4×C59, Dic3×C59, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(236, 2);