#72210663

order	2400
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, Sylow	C₅² ⋊ GU(2,𝔽₃)
derived	#6057⋊C₃ ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	#6057⋊C₃	C₄
Fitting, socle	C₅²	GU(2,𝔽₃)
prime radical, cosocle	((D₅⋊AGL(1,𝔽₅))⋊C₂)⋊C₃	C₂
LCS	G ▷ #6057⋊C₃
UCS	C₁

63 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3,2^3,5^2|a^2=b,b^2=f,d^2=f,e^2=f,c^a=c^2,d^a=de,d^c=e,e^a=ef,e^c=def,e^d=ef,g^a=g^3h^2,g^b=g^3,g^c=g^4h^2,g^d=g^3h^2,g^e=g^2,g^f=g^4,h^a=g^2h^2,h^b=h^3,h^c=g^2,h^d=h^2,h^e=g^4h^3,h^f=h^4

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(2,𝔽₅), C₂×#72210663
maximal normal subgroup of	C₂×#72210663
maximal quotient of	C₂×#72210663
automorphism group of	#6057

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