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Cayley diagram

C2×D43 #4301

order172
isoclinismisoclinic class of D43
automorphismAut(G) = C2×AGL(1,𝔽43) (order 3612) · Inn(G) = D43 · Out(G) = C2×C3×C7
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD86
direct productC2×D43
unambiguous semidirect productC43⋊C22
unambiguous semidirect product(C2×C43)⋊C2

Splittings

Fitting, socleC2×C43 ⋊ C2
derived, SylowC43 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2D43
commutator, cosocle, hypocenterC43C22
FrattiniC1G
Fitting, socleC2×C43C2
LCSG ▷ C43
UCSC1 ◁ C2

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 86

A minimal faithful permutation representation — 2 generators on 45 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);(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×D43, D172, C22⋊D43, …
maximal normal subgroup ofC4×D43, D172, C22⋊D43, …
maximal quotient ofDic86, C4×D43, D172, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(172, 3);