#72342952

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

Aliases

No aliases for this group.

Splittings

Fitting

C₂²×#7389 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂	#72342753
commutator	A₄×C₁₆	C₂²×C₈
Frattini	#346	C₂²×S₄
Fitting	C₂²×#7389	D₃
socle	C₂³	#10707069
prime radical	#72342799	C₄
cosocle	A₄×#346	C₂³
hypercenter	#7389	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 ◁ #346 ◁ #7389

1442 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)(5,7)(6,13)(8,24)(9,28)(10,18)(11,20)(12,29)(14,33)(15,23)(16,25)(17,22)(19,32)(21,35)(26,34)(27,30)(31,36);(2,4)(5,6,8,12,9,16,23,20,10,14,26,30,27,34,33,18,11,15,25,28,29,24,13,7,17,19,31,21,35,36,32,22);(6,12,14,21,15,28,19,30)(7,16,18,34,20,24,22,36)(8,23,25,13)(9,27)(10,17)(26,32,31,33)(29,35)

Polycyclic presentation:

2^9,3,2^2|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,h^2=i,b^a=be,c^a=cegh,c^b=ce,d^b=dhi,d^c=dh,e^a=eg,e^c=eghi,e^d=ei,f^b=fi,f^c=fi,g^a=gh,g^c=ghi,h^a=hi,h^c=hi,j^b=j^2,k^b=kl,k^j=l,l^j=kl

Conjugacy classes & character table

128 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, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.7*F.8);;
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.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
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.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
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);