#72276708

order	5376
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#1781 ⋊ C₂×C₃
Hall	#13385378 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #23900
Sylow	C₇ ⋊ C₃×#23900

Subgroups

	subgroup	G/H
center	C₂²	C₂×#6445⋊C₃
commutator, cosocle	C₂⁴×C₇	C₂⁴×C₃
Frattini	C₂⁴	C₂³×AGL(1,𝔽₇)
Fitting	C₇×#1781	C₂×C₃
socle	C₂²×C₇	C₂×C₃×#120
hypercenter	#1781	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#23900
LCS	G ▷ C₂⁴×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #1781

7040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 112

Polycyclic presentation:

2^4,3,2^4,7|a^2=f,c^2=fgi,d^2=fhi,b^a=bfh,c^a=cg,c^b=cfh,d^a=dghi,d^b=dfh,d^c=dgh,f^c=fi,f^d=fi,g^a=gi,g^c=gi,g^d=gi,h^c=hi,h^d=hi,j^d=j^6,j^e=j^4

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#23900
maximal normal subgroup of	AGL(1,𝔽₇)×#23900

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