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Cayley diagram

Dic58 #6600

order232
isoclinismisoclinic class of Dic58
automorphismAut(G) = D4×AGL(1,𝔽29) (order 6496) · Inn(G) = C2×D29 · Out(G) = C2×C4×C7
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic58
unambiguous semidirect productC29⋊Q8

Splittings

SylowC29 ⋊ Q8

Subgroups

subgroupG/H
center, FrattiniC2C2×D29
commutator, socle, prime radical, cosocleC2×C29C22
FittingC4×C29C2
hypercenterC4D29
hypocenterC29Q8
LCSG ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C4

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 232

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)

Polycyclic presentation:

2^3,29|a^2=c,b^2=c,b^a=bc,d^a=d^28

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#90773, Dic116, #90783, …
maximal normal subgroup of#90773, Dic116, #90783, …
maximal quotient of#90779, C4⋊Dic29, C2×Dic58, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(232, 4);