#72342940

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

Aliases

No aliases for this group.

Splittings

Fitting

C₂²×#12067 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂	#72342761
commutator	A₄×C₁₆	C₂²×C₈
Frattini	#346	C₂²×S₄
Fitting	C₂²×#12067	D₃
socle	C₂³	#10707070
prime radical	#72342752	C₄
cosocle	A₄×#346	C₂³
hypercenter	#12067	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 ◁ #12067

1581 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:

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

Polycyclic presentation:

2^9,3,2^2|a^2=d,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bg,c^a=ce,c^b=ce,d^b=di,d^c=dhi,e^a=eg,e^b=eghi,e^c=eghi,e^d=ei,f^c=fi,g^a=gh,g^b=ghi,g^c=ghi,h^a=hi,h^b=hi,h^c=hi,j^c=j^2,k^c=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, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
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.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*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, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> SetConjugate(coll, 11, 3, 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);