#72341940

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

#72281044 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×#1020⋊C₂
commutator	#5950	C₂³×C₄
Frattini	C₂³×C₄	C₂³×S₄
Fitting	#72281044	D₃
prime radical	C₂×#72236904	C₂
cosocle	C₂×#5950	C₂⁴
hypercenter	#102	#1020⋊C₂
hypocenter	#997	C₂×C₄×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

At least 10390 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^6,3,2^5|a^2=e,b^2=e,d^2=e,f^2=l,h^2=l,i^2=l,c^a=cl,c^b=cf,d^c=dl,f^b=fl,f^c=fl,g^d=g^2,h^a=hjl,h^b=hjl,h^c=hjl,h^d=hi,h^g=i,i^a=ijkl,i^b=ijkl,i^c=ijkl,i^d=il,i^g=hil,i^h=il,j^d=k,j^g=jk,j^i=jl,k^d=j,k^g=j,k^h=kl,k^i=kl

Conjugacy classes & character table

146 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, 1, F.5);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.12);;
gap> SetPower(coll, 8, F.12);;
gap> SetPower(coll, 9, F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.12);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.12);;
gap> SetConjugate(coll, 6, 3, F.6*F.12);;
gap> SetConjugate(coll, 7, 4, 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.10*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 9, 4, F.9*F.12);;
gap> SetConjugate(coll, 9, 7, F.8*F.9*F.12);;
gap> SetConjugate(coll, 9, 8, F.9*F.12);;
gap> SetConjugate(coll, 10, 4, F.11);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> SetConjugate(coll, 10, 9, F.10*F.12);;
gap> SetConjugate(coll, 11, 4, F.10);;
gap> SetConjugate(coll, 11, 7, F.10);;
gap> SetConjugate(coll, 11, 8, F.11*F.12);;
gap> SetConjugate(coll, 11, 9, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);