#72213412

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×D₄ ⋊ C₂×C₄
Sylow	C₅² ⋊ C₄×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂³×D₅⋊AGL(1,𝔽₅)
commutator, socle	C₂×C₅²	C₂⁴×C₄
Fitting	C₂×C₅²×D₄	C₂×C₄
prime radical	#13107821	C₂
cosocle	C₂×C₅⋊D₅	C₂⁵
hypercenter	C₂×D₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×D₄

3746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213584, #72213689, C₂×#72213412, …
maximal normal subgroup of	#72213584, #72213689, C₂×#72213412, …
maximal quotient of	C₂×#72213412, #72221567, #72221687, …
holomorph of	C₄×D₅, C₂²⋊D₅

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