#72222912

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×D₄ ⋊ SD₁₆
Sylow	C₃² ⋊ #62243

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂³×AΓL(1,𝔽₉)
commutator, cosocle	C₂×#157	C₂⁵
Fitting	C₂×C₃²×D₄	SD₁₆
socle	C₂×C₃²	C₂³×SD₁₆
prime radical	#11905868	C₂
hypercenter	C₂×D₄	AΓL(1,𝔽₉)
hypocenter	C₃²	#62243
LCS	G ▷ C₂×#157 ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×D₄

2591 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,3^2|b^2=g,e^2=fh,g^2=h,c^a=cf,d^a=df,e^a=ef,e^b=egh,e^c=ef,g^e=gh,i^b=i^2j^2,i^e=ij,i^g=i^2j,i^h=i^2,j^b=i^2,j^e=ij^2,j^g=ij,j^h=j^2

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	H(5,𝔽₂)×AΓL(1,𝔽₉), #72253254, #72253263
maximal normal subgroup of	H(5,𝔽₂)×AΓL(1,𝔽₉), #72253254, #72253263

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