#72227464

order	3584
automorphism	?
homology	Abelianization (H₁) = C₇ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

H(3,𝔽₈) ⋊ C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	H(3,𝔽₈)	C₇
Frattini, socle	C₂³	#90632
LCS	G ▷ H(3,𝔽₈)
UCS	C₁

710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 64

Polycyclic presentation:

7,2^9|b^a=bd,c^a=d,d^a=bcd,e^a=ef,e^b=eh,e^c=ehi,e^d=ej,f^a=g,f^b=fhj,f^c=fj,f^d=fij,g^a=eij,g^b=gij,g^c=gi,g^d=gh,h^a=ij,i^a=j,j^a=h

Conjugacy classes & character table

17 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227465
maximal quotient of	#72227961

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