#72241968

order	8112
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃×C₁₃² ⋊ C₄²
Sylow	C₁₃² ⋊ C₄×Dic₃
Sylow	C₃ ⋊ #261²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₁₃²	C₄²
prime radical, cosocle	C₃×D₁₃²	C₂²
LCS	G ▷ C₃×C₁₃²
UCS	C₁

138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 156

Polycyclic presentation:

2^4,3,13^2|a^2=c,b^2=d,e^a=e^2,f^a=f^8,f^b=f^8,f^c=f^12,f^d=f^12,g^a=f^6g,g^b=g^8,g^c=f^2g,g^d=g^12

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#261², #72242001
maximal normal subgroup of	D₃×#261², #72242001

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 13, 13]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^8);;
gap> SetConjugate(coll, 6, 2, F.6^8);;
gap> SetConjugate(coll, 6, 3, F.6^12);;
gap> SetConjugate(coll, 6, 4, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.6^6*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^8);;
gap> SetConjugate(coll, 7, 3, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 4, F.7^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);