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Cayley diagram

Dic47 #4420

order188
isoclinismisoclinic class of D47
automorphismAut(G) = C2×AGL(1,𝔽47) (order 4324) · Inn(G) = D47 · Out(G) = C2×C23
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic47
unambiguous semidirect productC47⋊C4

Splittings

derived, SylowC47 ⋊ C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D47
commutator, hypocenterC47C4
Fitting, socle, prime radical, cosocleC2×C47C2
LCSG ▷ C47
UCSC1 ◁ C2

6 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);(1,3,2,4)(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:

2^2,47|a^2=b,c^a=c^46

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic94, C4×D47, C2×Dic47, …
maximal normal subgroup ofDic94, C4×D47, C2×Dic47, …
maximal quotient ofC47⋊C8, C2×Dic47, C3×Dic47, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(188, 1);