#72259342

order	8112
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Hall	#72259197 ⋊ C₃
Hall	#92788 ⋊ M₁₆
Sylow	C₁₃² ⋊ C₃×M₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#14523146
commutator, socle	C₂×C₁₃²	C₂×C₃×C₄
Fitting	C₄×C₁₃²	C₃×C₄
prime radical	C₂×#11759815	C₄
cosocle	#10621725	C₂²×C₃
hypercenter	C₄	#14523146
hypocenter	C₁₃²	C₃×M₁₆
LCS	G ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₄

176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 104

Polycyclic presentation:

2^2,3,2^2,13^2|a^2=d,b^2=e,d^2=e,b^a=be,f^a=f^5,f^c=f^3,f^d=f^12,g^a=f^6g^8,g^c=g^3,g^d=g^12

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259343
maximal normal subgroup of	#72259343

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