#72259667

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₇×H(5,𝔽₂) ⋊ C₂×C₃
Hall	#11744345 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #3585
Sylow	C₇ ⋊ C₃×#3585

Subgroups

	subgroup	G/H
center	C₂	C₂³×(C₂²⋊D₇)⋊C₃
commutator, cosocle	C₄×C₇	C₂⁵×C₃
Frattini	C₄	C₂⁴×AGL(1,𝔽₇)
Fitting	C₂×C₇×H(5,𝔽₂)	C₂×C₃
socle	C₂×C₇	C₂³×C₃×D₄
hypercenter	C₂×H(5,𝔽₂)	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#3585
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

2928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 112

Polycyclic presentation:

2^5,3,2^2,7|a^2=h,g^2=h,b^a=bg,c^b=cgh,d^a=dh,e^a=eh,e^b=eh,g^a=gh,g^b=gh,g^c=gh,i^b=i^6,i^f=i^2

Conjugacy classes & character table

157 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#3585
maximal normal subgroup of	AGL(1,𝔽₇)×#3585

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