#72254457

order	8064
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₇×H(5,𝔽₂) ⋊ C₂²×C₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ C₂×#453
Sylow	C₇ ⋊ C₃×D₃×#453

Subgroups

	subgroup	G/H
center	C₂	#72244206
commutator, cosocle	C₂×C₃×C₄×C₇	C₂⁴×C₃
Frattini	C₂×C₄	C₂²×D₃×AGL(1,𝔽₇)
Fitting	C₃×C₇×H(5,𝔽₂)	C₂²×C₃
socle	C₂×C₃×C₇	C₂×C₃×#120
hypercenter	H(5,𝔽₂)	D₃×AGL(1,𝔽₇)
hypocenter	C₃×C₇	C₂×C₃×#453
LCS	G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ H(5,𝔽₂)

3544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^4,3,2^3,3,7|g^2=h,b^a=bg,d^a=dh,d^b=df,f^a=fh,g^a=gh,g^b=gh,g^d=gh,i^c=i^2,j^b=j^6,j^c=j^6,j^e=j^4

Conjugacy classes & character table

177 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×AGL(1,𝔽₇)×#453
maximal normal subgroup of	D₃×AGL(1,𝔽₇)×#453

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