#72277859

order	2400
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

solvable

Aliases

No aliases for this group.

Splittings

derived	#6057 ⋊ C₃×C₄
Hall	#11717308 ⋊ C₃
Sylow	C₅² ⋊ C₄×SL(2,𝔽₃)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	((D₅⋊AGL(1,𝔽₅))⋊C₂)⋊C₃
commutator, hypocenter	#6057	C₃×C₄
Fitting, socle	C₂×C₅²	#41⋊C₃
prime radical	C₂×#6057⋊C₃	C₂
cosocle	C₂×#6057	C₂×C₃
LCS	G ▷ #6057
UCS	C₁ ◁ C₂

75 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 100

Polycyclic presentation:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277865, C₄×((D₅⋊AGL(1,𝔽₅))⋊C₂)⋊C₃
maximal normal subgroup of	C₄×((D₅⋊AGL(1,𝔽₅))⋊C₂)⋊C₃

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