#72213282

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×D₄ ⋊ C₂×C₄
Sylow	C₅² ⋊ C₄×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	C₂×#11717375
commutator	C₄×C₅²	C₂³×C₄
Frattini	C₄	C₂²×D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₅²×D₄	C₂×C₄
socle	C₂×C₅²	C₂×C₄×D₄
prime radical	#13106298	C₂
cosocle	C₄×C₅⋊D₅	C₂⁴
hypercenter	C₂×D₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

1522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213584, #72221154, C₂×#72213282, …
maximal normal subgroup of	#72213584, #72221154, C₂×#72213282, …
maximal quotient of	#72220815, C₂×#72213282, #72223303, …
holomorph of	#168

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