Finite Group Databasebeta

← Prev (#92446) Next (#92448) →

Cayley diagram

Dic123 #92447

order492
isoclinismisoclinic class of D123
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic123
unambiguous semidirect productC3⋊Dic41

Splittings

derived, HallC3×C41 ⋊ C4
SylowC41 ⋊ Dic3
SylowC3 ⋊ Dic41

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D123
commutator, hypocenterC3×C41C4
Fitting, socle, prime radical, cosocleC2×C3×C41C2
LCSG ▷ C3×C41
UCSC1 ◁ C2

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 48
transitive degreeμt(G) = 492

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48);(1,3,2,4)(6,7)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)

Polycyclic presentation:

2^2,3,41|a^2=b,c^a=c^2,d^a=d^40

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic3×D41, D3×Dic41, #11758296, …
maximal normal subgroup ofDic3×D41, D3×Dic41, #11758296, …
maximal quotient ofC3⋊#66491, C2×Dic123, Dic369, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(492, 3);