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Cayley diagram

C22×C67 #63181

order268
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C3×D3×C11 (order 396) · Inn(G) = C1 · Out(G) = C2×C3×D3×C11
homologyAbelianization (H1) = G · Schur multiplier (H2) = C2

Aliases

abelianC22×C67

Subgroups

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

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 71 points:

(1,2)(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);(3,4)

Conjugacy classes & character table

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

Relationships

center ofC22⋊C4×C67, C4⋊C4×C67, C2×D4×C67, …
derived subgroup ofC22⋊(C67⋊C3), A4×D67, D67⋊A4, …
Fitting subgroup ofC2×Dic67, C22⋊D67, C22×D67, …
socle ofC2×Dic67, C2×C4×C67, C22×D67, …
maximal subgroup ofC2×Dic67, C22⋊D67, C2×C4×C67, …
maximal normal subgroup ofC2×Dic67, C22⋊D67, C2×C4×C67, …
maximal quotient ofC2×C4×C67, D4×C67, Q8×C67, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(268, 4);