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Cayley diagram

D67 #3615

order134
isoclinismisoclinic class of D67
automorphismAut(G) = AGL(1,𝔽67) (order 4422) · Inn(G) = G · Out(G) = C3×C11
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD67
unambiguous semidirect productC67⋊C2

Splittings

Fitting, socle, derived, SylowC67 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC67C2
LCSG ▷ C67
UCSC1

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 67
transitive degreeμt(G) = 67

A minimal faithful permutation representation — 2 generators on 67 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);(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)

Polycyclic presentation:

2,67|b^a=b^66

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D67, D67⋊C3, C3×D67, …
maximal normal subgroup ofC2×D67, D67⋊C3, C3×D67, …
maximal quotient ofDic67, C2×D67, C3×D67, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(134, 1);