#72242199

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #1368

Subgroups

	subgroup	G/H
center	C₂²	#11716861
commutator	C₅²×C₈	C₂×C₈
Frattini	C₂×C₈	C₂×#1043
Fitting	C₂×C₅²×C₁₆	C₄
socle, prime radical	C₂²×C₅²	#106
cosocle	C₈×#1039	C₂²
hypercenter	C₂×C₁₆	#1043
hypocenter	C₅²	#1368
LCS	G ▷ C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 3200

Polycyclic presentation:

2^7,5^2|a^2=c,b^2=d,c^2=e,d^2=f,f^2=g,b^a=bd,c^b=cg,d^a=df,f^a=fg,h^a=h^3,h^b=h^4,h^c=h^4,i^a=i^3,i^b=i^4,i^c=i^4

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242207, #72253157, #72253161
maximal normal subgroup of	#72242207, #72253157, #72253161

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