#72215818

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting

#10586999 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×D₄×#1020
commutator, cosocle	#5950	C₂⁴
Frattini	C₄	C₂³×#1020
Fitting	#10586999	D₃
hypercenter	AGL(1,ℤ/8)	#1020
hypocenter	#997	C₂²×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

At least 10862 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^5,3,2^5|a^2=e,e^2=k,b^a=bek,c^a=ck,e^b=ek,f^d=f^2,g^d=gh,g^f=h,h^f=gh,i^d=gjk,i^f=j,i^g=ik,i^h=ik,j^d=ghik,j^f=ij,j^h=jk

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72245616, #72252511, C₂×#72215818
maximal normal subgroup of	#72245616, #72252511, C₂×#72215818
maximal quotient of	C₂×#72215818
holomorph of	#131

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