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Cayley diagram

D188 #67339

order376
isoclinismisoclinic class of Dic94
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD188
unambiguous semidirect productC4⋊D47

Splittings

FittingC4×C47 ⋊ C2
SylowC47 ⋊ D4

Subgroups

subgroupG/H
center, FrattiniC2C2×D47
commutator, socle, cosocleC2×C47C22
FittingC4×C47C2
hypercenterC4D47
hypocenterC47D4
LCSG ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C4

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 51
transitive degreeμt(G) = 188

A minimal faithful permutation representation — 2 generators on 51 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);(2,4)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)

Polycyclic presentation:

2^3,47|b^2=c,b^a=bc,d^a=d^46

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10625325, D376, #10625334, …
maximal normal subgroup of#10625325, D376, #10625334, …
maximal quotient of#10625325, D376, Dic188, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(376, 5);