#72220092

order	31104
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	#127×#6900 ⋊ C₂²
Sylow	#6900 ⋊ C₂×#453

Subgroups

	subgroup	G/H
center	C₂	#72220091
commutator, cosocle	C₂×C₄×C₉²	C₂⁴×C₃
Frattini	C₂×C₃²×C₄	C₂²×C₃×D₃²
Fitting	#127×#6900	C₂²
socle	C₂×C₃²	C₃×#10595278
hypercenter	#127	#92265⋊C₂
hypocenter	C₉²	C₂×C₃×#453
LCS	G ▷ C₂×C₄×C₉² ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #127

11748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3,2^3,3^4|a^2=h,b^2=f,g^2=h,i^3=k,j^3=l,b^a=bg,c^b=cf,f^a=fh,g^a=gh,g^c=gh,i^a=i^2k^2,i^d=i^2k^2,i^e=ik^2,j^a=i^2j,j^d=j^2l^2,j^e=jl^2,k^a=k^2,k^d=k^2,l^a=k^2l,l^d=l^2

Conjugacy classes & character table

336 conjugacy classes — view the full character table.

Relationships

holomorph of

Dic₁₈, D₃₆

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