#72277078

order	5376
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₇×AΣL(1,𝔽₁₆) ⋊ C₂×C₃
Hall	#13272735 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #13343
Sylow	C₇ ⋊ C₃×#13343

Subgroups

	subgroup	G/H
center	C₂²	#89513⋊C₃
commutator, cosocle	C₂²×C₇×D₄	C₂³×C₃
Frattini	C₂²×D₄	C₂²×AGL(1,𝔽₇)
Fitting	C₂×C₇×AΣL(1,𝔽₁₆)	C₂×C₃
socle	C₂²×C₇	C₃×B(4,𝔽₂)
hypercenter	C₂×AΣL(1,𝔽₁₆)	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#13343
LCS	G ▷ C₂²×C₇×D₄ ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×AΣL(1,𝔽₁₆)

2168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 19 points:

(2,4)(6,9,10,7,8,11)(12,14,16,13)(15,19,18,17);(5,6,8,11,10,9,7)(12,13,15,17)(14,18,19,16);(1,2)(3,4)(13,14)

Polycyclic presentation:

2^3,3,2^5,7|b^2=e,c^2=i,b^a=bf,c^a=cfg,c^b=cei,e^a=eh,f^b=fh,f^c=fi,f^e=fi,g^b=gi,g^c=gi,h^b=hi,h^c=hi,j^c=j^6,j^d=j^2

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#13343
maximal normal subgroup of	AGL(1,𝔽₇)×#13343

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