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Cayley diagram

Dic43 #4299

order172
isoclinismisoclinic class of D43
automorphismAut(G) = C2×AGL(1,𝔽43) (order 3612) · Inn(G) = D43 · Out(G) = C2×C3×C7
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic43
unambiguous semidirect productC43⋊C4

Splittings

derived, SylowC43 ⋊ C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D43
commutator, hypocenterC43C4
Fitting, socle, prime radical, cosocleC2×C43C2
LCSG ▷ C43
UCSC1 ◁ C2

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

2^2,43|a^2=b,c^a=c^42

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic86, C4×D43, C2×Dic43, …
maximal normal subgroup ofDic86, C4×D43, C2×Dic43, …
maximal quotient ofC43⋊C8, C2×Dic43, C3×Dic43, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(172, 1);