#72277358

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₇×B(4,𝔽₂) ⋊ C₂×C₃
Hall	#13272733 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #13343
Sylow	C₇ ⋊ C₃×#13343

Subgroups

	subgroup	G/H
center	C₂²	#89804⋊C₃
commutator, cosocle	C₂²×C₇×D₄	C₂³×C₃
Frattini	C₂²×D₄	C₂²×AGL(1,𝔽₇)
Fitting	C₂×C₇×B(4,𝔽₂)	C₂×C₃
socle	C₂²×C₇	C₃×B(4,𝔽₂)
hypercenter	C₂×B(4,𝔽₂)	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#13343
LCS	G ▷ C₂²×C₇×D₄ ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×B(4,𝔽₂)

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

125 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, 3, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.10^6);;
gap> SetConjugate(coll, 10, 4, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);