#72222866

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×M₃₂ ⋊ C₂²
Sylow	C₃² ⋊ C₂×#2148

Subgroups

	subgroup	G/H
center	C₄	#10590940
commutator	C₃²×C₈	C₂³×C₄
Frattini	C₂×C₈	C₂²×D₃²
Fitting	C₂×C₃²×M₃₂	C₂²
socle	C₂×C₃²	C₂²×#102
prime radical	C₂×#10594684	C₂
cosocle	C₂×C₃²×C₈	C₂⁴
hypercenter	C₂×M₃₂	D₃²
hypocenter	C₃²	C₂×#2148
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

2508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,3^2|a^2=e,e^2=gh,f^2=g,g^2=h,b^a=bf,c^a=ch,c^b=ch,e^b=eg,f^a=fh,f^b=fgh,g^b=gh,i^b=i^2,i^d=i^2,j^b=i^2j,j^d=j^2

Conjugacy classes & character table

168 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 3, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> SetConjugate(coll, 9, 4, F.9^2);;
gap> SetConjugate(coll, 10, 2, F.9^2*F.10);;
gap> SetConjugate(coll, 10, 4, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);