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Cayley diagram

#3556⋊C3 #85661

order384
isoclinismisoclinic class of #3556⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C24

Aliases

unambiguous semidirect product#3556⋊C3

Splittings

Fitting, Sylow#3556 ⋊ C3
derived#564 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#564C2×C3
Frattini, socleC22C2×C22⋊A4
Fitting#3556C3
LCSG ▷ #564
UCSC1

175 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2,3,2^6|c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bh,c^a=cgh,d^a=dg,c^b=defg,d^b=cdfgh,e^b=f,f^b=efh,g^b=h,h^b=gh,d^c=dg,e^c=egh,f^c=fh,e^d=eh,f^d=fg

Conjugacy classes & character table

18 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#8533⋊C3, #19083⋊C3, C2×#3556⋊C3, …
maximal normal subgroup of#8533⋊C3, #19083⋊C3, C2×#3556⋊C3, …
maximal quotient of#34526⋊C3, #37613⋊C3, #37643⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18237);