#72259641

order	16224
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	D₄×C₁₃² ⋊ C₂²×C₃
Hall	#72259284 ⋊ C₃
Hall	#92788 ⋊ H(5,𝔽₂)
Sylow	C₁₃² ⋊ C₃×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×#11759815⋊C₂
commutator, socle, cosocle	C₂×C₁₃²	C₂⁴×C₃
Fitting	D₄×C₁₃²	C₂²×C₃
hypercenter	D₄	#11759815⋊C₂
hypocenter	C₁₃²	C₃×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ D₄

988 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 104

Polycyclic presentation:

2^4,3,2,13^2|b^a=bf,c^a=cf,d^b=df,g^a=g^12,g^c=g^12,g^e=g^9,h^a=g^2h,h^c=h^12,h^e=h^9

Conjugacy classes & character table

151 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259728
maximal normal subgroup of	#72259728

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, 2, 2, 3, 2, 13, 13]);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^12);;
gap> SetConjugate(coll, 7, 3, F.7^12);;
gap> SetConjugate(coll, 7, 5, F.7^9);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^12);;
gap> SetConjugate(coll, 8, 5, F.8^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);