#72342996

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting

C₂²×#12068 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂	#72342761
commutator	A₄×C₁₆	C₂²×C₈
Frattini	#346	C₂²×S₄
Fitting	C₂²×#12068	D₃
socle	C₂³	#10707070
prime radical	#72342799	C₄
cosocle	A₄×#346	C₂³
hypercenter	#12068	S₄
hypocenter	A₄	AGL(1,ℤ/32)
LCS	G ▷ A₄×C₁₆ ▷ C₈×A₄ ▷ C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1313 ◁ #12068

1537 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9,3,2^2|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,g^2=h,h^2=i,b^a=bghi,c^a=ce,c^b=ce,d^b=dhi,d^c=dg,e^a=ehi,e^c=eghi,e^d=ei,f^b=fi,f^c=fh,g^a=gi,g^c=ghi,h^c=hi,j^a=j^2,j^b=j^2,k^a=kl,k^b=kl,k^j=l,l^j=kl

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×AGL(1,ℤ/32)
maximal normal subgroup of	S₄×AGL(1,ℤ/32)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> SetConjugate(coll, 11, 10, F.12);;
gap> SetConjugate(coll, 12, 10, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);