#72269320

order	6048
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Hall	C₃²×ΓL(1,𝔽₈) ⋊ AGL(1,ℤ/8)
Sylow	C₇ ⋊ C₃×#63937

Subgroups

	subgroup	G/H
center	C₂	#72222040
commutator, Fitting, cosocle	C₃²×C₄×C₇	C₂³×C₃
Frattini, hypercenter	C₄	D₃²×AGL(1,𝔽₇)
socle	C₂×C₃²×C₇	C₂×C₃×D₄
hypocenter	C₃²×C₇	C₃×AGL(1,ℤ/8)
LCS	G ▷ C₃²×C₄×C₇ ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂ ◁ C₄

768 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 168

Polycyclic presentation:

2^3,3,2^2,3^2,7|a^2=e,e^2=f,b^a=bef,c^a=cf,e^b=ef,g^a=g^2,g^b=g^2,g^c=g^2,h^a=g^2h,h^b=h^2,h^c=h^2,i^c=i^6,i^d=i^4

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72269327
maximal normal subgroup of	#72269327

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