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Cayley diagram

Dic62 #6935

order248
isoclinismisoclinic class of Dic62
automorphismAut(G) = D4×AGL(1,𝔽31) (order 7440) · Inn(G) = C2×D31 · Out(G) = C22×C3×C5
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic62
unambiguous semidirect productC31⋊Q8

Splittings

SylowC31 ⋊ Q8

Subgroups

subgroupG/H
center, FrattiniC2C2×D31
commutator, socle, prime radical, cosocleC2×C31C22
FittingC4×C31C2
hypercenterC4D31
hypocenterC31Q8
LCSG ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C4

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 39
transitive degreeμt(G) = 248

A minimal faithful permutation representation — 2 generators on 39 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39);(1,5,3,7)(2,6,4,8)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)

Polycyclic presentation:

2^3,31|a^2=c,b^2=c,b^a=bc,d^a=d^30

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#92470, Dic124, #92480, …
maximal normal subgroup of#92470, Dic124, #92480, …
maximal quotient ofDic31⋊C4, C4⋊Dic31, C2×Dic62, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(248, 3);