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Cayley diagram

C4×C53 #6156

order212
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C4×C13 (order 104) · Inn(G) = C1 · Out(G) = C2×C4×C13
holomorphG ⋊ Aut(G) = D4×AGL(1,𝔽53)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC212
abelianC4×C53

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2C2×C53
socle, prime radicalC2×C53C2
LCSG ▷ C1
UCSC1 ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 57
transitive degreeμt(G) = 212

A minimal faithful permutation representation — 1 generator on 57 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)

Conjugacy classes & character table

212 conjugacy classes — view the full character table.

Relationships

center ofM16×C53, #41×C53, C3⋊C8×C53, …
derived subgroup of#11719913, D424, Dic212, …
Frattini subgroup ofDic11236, D22472
Fitting subgroup of#88322, Dic106, C4×D53, …
maximal subgroup of#88322, C8×C53, Dic106, …
maximal normal subgroup of#88322, C8×C53, Dic106, …
maximal quotient ofC8×C53, C2×C4×C53, Dic3×C53, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(212, 2);