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Cayley diagram

C3⋊#4223 #92450

order492
isoclinismisoclinic class of C3⋊#4223
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊#4223

Splittings

Fitting, socle, derived, HallC3×C41 ⋊ C4
SylowC41 ⋊ Dic3
SylowC3 ⋊ #4223

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C41C4
prime radical, cosocleC3×D41C2
LCSG ▷ C3×C41
UCSC1

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 44
transitive degreeμt(G) = 123

A minimal faithful permutation representation — 2 generators on 44 points:

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

Polycyclic presentation:

2^2,3,41|a^2=b,c^a=c^2,d^a=d^9,d^b=d^40

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#4223, C2×C3⋊#4223, C9⋊#4223, …
maximal normal subgroup ofD3×#4223, C2×C3⋊#4223, C3×C3⋊#4223, …
maximal quotient ofC3⋊#66493, C2×C3⋊#4223, C9⋊#4223, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(492, 6);