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Cayley diagram

Dic114 #90714

order456
isoclinismisoclinic class of Dic114
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic114

Splittings

HallC3×C19 ⋊ Q8
SylowC19 ⋊ Dic6
SylowC3 ⋊ Dic38

Subgroups

subgroupG/H
center, FrattiniC2C2×D57
commutator, socle, prime radical, cosocleC2×C3×C19C22
FittingC3×C4×C19C2
hypercenterC4D57
hypocenterC3×C19Q8
LCSG ▷ C2×C3×C19 ▷ C3×C19
UCSC1 ◁ C2 ◁ C4

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 456

A minimal faithful permutation representation — 2 generators on 30 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);(1,5,3,7)(2,6,4,8)(10,11)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)

Polycyclic presentation:

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

Conjugacy classes & character table

117 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11744823, #11744825, #11744828, …
maximal normal subgroup of#11744823, #11744825, #11744828, …
maximal quotient of#11744887, C4⋊Dic57, C2×Dic114, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 34);