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Cayley diagram

Dic38 #3886

order152
isoclinismisoclinic class of Dic38
automorphismAut(G) = D4×AGL(1,𝔽19) (order 2736) · Inn(G) = C2×D19 · Out(G) = C22×C9
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic38
unambiguous semidirect productC19⋊Q8

Splittings

SylowC19 ⋊ Q8

Subgroups

subgroupG/H
center, FrattiniC2C2×D19
commutator, socle, prime radical, cosocleC2×C19C22
FittingC4×C19C2
hypercenterC4D19
hypocenterC19Q8
LCSG ▷ C2×C19 ▷ C19
UCSC1 ◁ C2 ◁ C4

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 152

A minimal faithful permutation representation — 2 generators on 27 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);(1,5,3,7)(2,6,4,8)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)

Polycyclic presentation:

2^3,19|a^2=c,b^2=c,b^a=bc,d^a=d^18

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#64516, Dic76, #64526, …
maximal normal subgroup of#64516, Dic76, #64526, …
maximal quotient ofDic19⋊C4, C4⋊Dic19, C2×Dic38, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(152, 3);