#72274997

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

C₃×H(7,𝔽₂) ⋊ C₂×D₃

Subgroups

	subgroup	G/H
center	C₂	C₂²×#10593244
commutator, cosocle, hypocenter	C₃×#997	C₂⁴
Frattini	C₂³	C₂²×D₃×S₄
Fitting	C₃×H(7,𝔽₂)	C₂×D₃
socle	C₂×C₃	C₂²×#1020⋊C₂
hypercenter	D₄	#10593244
LCS	G ▷ C₃×#997
UCS	C₁ ◁ C₂ ◁ D₄

7555 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^4,3^2,2^5|a^2=k,g^2=k,h^2=k,b^a=bk,d^a=dk,e^d=e^2,f^c=f^2,g^c=gh,g^d=gjk,g^f=h,h^c=hk,h^d=hik,h^f=ghk,h^g=hk,i^f=ij,i^g=ik,j^c=ij,j^f=i,j^h=jk

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72217027
maximal normal subgroup of	D₃×#72217027

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