#72222774

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ C₂×#2110

Subgroups

	subgroup	G/H
center	C₈	C₄×#624
commutator	C₃²×C₄	C₂²×C₄²
Frattini	C₂×C₈	C₂²×D₃²
Fitting	C₂×C₃²×M₃₂	C₂²
socle	C₂×C₃²	C₂²×#97
prime radical	C₂³×C₃⋊D₃	M₁₆
cosocle	C₂×C₃²×C₈	C₂⁴
hypercenter	C₂×M₃₂	D₃²
hypocenter	C₃²	C₂×#2110
LCS	G ▷ C₃²×C₄ ▷ C₃²
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂

1676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,3^2|a^2=f,b^2=e,f^2=g,g^2=h,b^a=bgh,c^a=ch,c^b=ch,e^a=eh,f^b=fh,i^b=i^2,i^d=i^2,j^b=i^2j,j^d=j^2

Conjugacy classes & character table

216 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228223
maximal normal subgroup of	#72228223

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.6);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*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);