#72237983

order	3584
automorphism	?
homology	Abelianization (H₁) = C₇ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#8932885 ⋊ C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	#8932885	C₇
Frattini, socle	C₂³	#90631
LCS	G ▷ #8932885
UCS	C₁

1505 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has 1505 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 56

Polycyclic presentation:

7,2^9|b^2=h,c^2=ij,d^2=i,b^a=bc,c^a=dj,c^b=chi,d^a=bh,d^b=dj,d^c=dh,e^a=ef,e^b=ej,e^c=ej,e^d=ehi,f^a=g,f^b=fh,f^c=fi,g^a=ehi,g^c=gj,g^d=gh,h^a=j,i^a=h,j^a=hi

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

automorphism group of

#2104

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