#72229823

order	15876
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₂×C₃²

Aliases

No aliases for this group.

Splittings

Hall	#72224635 ⋊ C₂²
Hall	D₇² ⋊ C₃×3₋¹⁺²
Sylow	C₇² ⋊ C₂²×C₃×3₋¹⁺²

Subgroups

	subgroup	G/H
center, Frattini	C₃	C₃×AGL(1,𝔽₇)²
commutator, socle, cosocle	C₃×C₇²	C₂²×C₃³
Fitting	C₃²×C₇²	C₂²×C₃²
prime radical	C₃²×#64423⋊C₂	C₃
hypercenter	C₃²	AGL(1,𝔽₇)²
hypocenter	C₇²	C₂²×C₃×3₋¹⁺²
LCS	G ▷ C₃×C₇² ▷ C₇²
UCS	C₁ ◁ C₃ ◁ C₃²

816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 126

Polycyclic presentation:

2^2,3^4,7^2|c^3=f,d^c=df,g^a=g^6,g^b=g^6,g^c=g^4,g^e=g^2,h^a=g^6h,h^b=h^6,h^c=g^5h,h^e=h^2

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

Relationships

holomorph of

D₇⋊C₉

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