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Cayley diagram

Dic74 #64439

order296
isoclinismisoclinic class of Dic74
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic74
unambiguous semidirect productC37⋊Q8

Splittings

SylowC37 ⋊ Q8

Subgroups

subgroupG/H
center, FrattiniC2C2×D37
commutator, socle, prime radical, cosocleC2×C37C22
FittingC4×C37C2
hypercenterC4D37
hypocenterC37Q8
LCSG ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C4

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 296

A minimal faithful permutation representation — 2 generators on 45 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(1,5,3,7)(2,6,4,8)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)

Polycyclic presentation:

2^3,37|a^2=c,b^2=c,b^a=bc,d^a=d^36

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10596930, Dic148, #10596940, …
maximal normal subgroup of#10596930, Dic148, #10596940, …
maximal quotient of#10596936, C4⋊Dic37, C2×Dic74, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(296, 4);