#72222918

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting

#612424 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#1020⋊C₂
commutator, cosocle	C₂²×C₂²⋊A₄	C₂⁴
Frattini, socle	C₂⁴	C₂³×S₄
Fitting	#612424	D₃
hypercenter	C₂×D₄	#1020⋊C₂
hypocenter	C₂²⋊A₄	C₂×#120
LCS	G ▷ C₂²×C₂²⋊A₄ ▷ C₂²⋊A₄
UCS	C₁ ◁ C₂² ◁ C₂×D₄

At least 8213 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^6,3,2^4|b^a=bf,c^b=ce,d^a=df,d^c=de,g^d=g^2,h^a=hj,h^d=hi,h^g=i,i^a=ik,i^g=hi,j^d=jk,j^g=k,k^g=jk

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72222918
maximal normal subgroup of	C₂×#72222918
maximal quotient of	#72222995, #72256496, #72341433, …
outer automorphism group of	C₂×C₄×Q₈

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