#72343911

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting

#72312259 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×#1020⋊C₂
commutator, cosocle	C₂×#5950	C₂⁴
Frattini	C₂³×C₄	C₂³×S₄
Fitting	#72312259	D₃
hypercenter	C₂×SD₁₆	#1020⋊C₂
hypocenter	#997	C₂×#120
LCS	G ▷ C₂×#5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

At least 15384 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^6,3,2^5|d^2=f,f^2=l,d^a=df,d^b=de,f^a=fl,g^c=g^2,h^a=hjl,h^b=hjl,h^c=hijkl,h^d=hjl,h^g=i,i^a=ikl,i^b=ikl,i^c=ikl,i^d=ikl,i^g=hi,j^c=jk,j^g=k,j^i=jl,k^g=jk,k^h=kl

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

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, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.12);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6*F.12);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.12);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.10*F.12);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 4, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 7, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> SetConjugate(coll, 10, 7, F.11);;
gap> SetConjugate(coll, 10, 9, F.10*F.12);;
gap> SetConjugate(coll, 11, 7, F.10*F.11);;
gap> SetConjugate(coll, 11, 8, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);