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Cayley diagram

D172 #66810

order344
isoclinismisoclinic class of Dic86
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD172
unambiguous semidirect productC4⋊D43

Splittings

FittingC4×C43 ⋊ C2
SylowC43 ⋊ D4

Subgroups

subgroupG/H
center, FrattiniC2C2×D43
commutator, socle, cosocleC2×C43C22
FittingC4×C43C2
hypercenterC4D43
hypocenterC43D4
LCSG ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C4

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 47
transitive degreeμt(G) = 172

A minimal faithful permutation representation — 2 generators on 47 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);(2,4)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)

Polycyclic presentation:

2^3,43|b^2=c,b^a=bc,d^a=d^42

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10621873, D344, #10621882, …
maximal normal subgroup of#10621873, D344, #10621882, …
maximal quotient of#10621873, D344, Dic172, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(344, 5);