#72254408

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₇×C₂²⋊C₄ ⋊ C₂²×C₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ C₂×#99
Sylow	C₇ ⋊ C₃×D₃×#99

Subgroups

	subgroup	G/H
center	C₂	#72244188
commutator	C₂²×C₃×C₇	C₂²×C₃×C₄
Frattini	C₂³	C₂×D₃×AGL(1,𝔽₇)
Fitting	C₃×C₇×C₂²⋊C₄	C₂²×C₃
socle	C₂×C₃×C₇	C₂×C₃×C₂²⋊C₄
prime radical	C₂×#66694⋊C₃	C₂
cosocle	C₂³×C₃×C₇	C₂³×C₃
hypercenter	C₂²⋊C₄	D₃×AGL(1,𝔽₇)
hypocenter	C₃×C₇	C₂×C₃×#99
LCS	G ▷ C₂²×C₃×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

892 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^3,3,2^3,3,7|a^2=f,b^a=be,c^a=ce,e^a=eg,f^b=fg,f^c=fg,h^b=h^2,i^c=i^6,i^d=i^4

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#99×AGL(1,𝔽₇), #72269452, #72269461
maximal normal subgroup of	D₃×#99×AGL(1,𝔽₇), #72269452, #72269461

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, 2, 3, 7]);;
gap> SetPower(coll, 1, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, 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);