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Cayley diagram

C4×C67 #63179

order268
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C22×C3×C11 (order 132) · Inn(G) = C1 · Out(G) = C22×C3×C11
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC268
abelianC4×C67
multiplicative(ℤ/269)ˣ, (ℤ/538)ˣ

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2C2×C67
socle, prime radicalC2×C67C2
LCSG ▷ C1
UCSC1 ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 71
transitive degreeμt(G) = 268

A minimal faithful permutation representation — 1 generator on 71 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)

Conjugacy classes & character table

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

Relationships

center ofM16×C67, #41×C67, C3⋊C8×C67, …
derived subgroup of#11761613, D536, #11761615, …
Fitting subgroup ofC67⋊C8, Dic134, C4×D67, …
maximal subgroup ofC67⋊C8, C8×C67, Dic134, …
maximal normal subgroup ofC67⋊C8, C8×C67, Dic134, …
maximal quotient ofC8×C67, C2×C4×C67, Dic3×C67, …
automorphism group ofC269, C2×C269
outer automorphism group ofC269, C2×C269

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(268, 2);