#72224650

order	14400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃²×C₅² ⋊ C₂²×C₄
Hall	C₃²×C₅² ⋊ C₂×C₄×D₄
Sylow	C₅² ⋊ C₄×#3811
Sylow	C₃² ⋊ C₂×#11717375

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#72210392
commutator, socle	C₂×C₃²×C₅²	C₂³×C₄
Fitting	C₂²×C₃²×C₅²	C₂²×C₄
prime radical	#72224623	C₂
cosocle	C₂×C₃²×C₅⋊D₅	C₂⁴
hypercenter	C₂²	#72210392
hypocenter	C₃²×C₅²	C₂×C₄×D₄
LCS	G ▷ C₂×C₃²×C₅² ▷ C₃²×C₅²
UCS	C₁ ◁ C₂ ◁ C₂²

5296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 4 generators on 20 points:

(6,7,8,9)(10,11,12,13)(15,16,17)(18,19,20);(5,6,9,7,8)(16,17)(19,20);(2,4);(1,2,3,4)(11,12)(13,14)(19,20)

Polycyclic presentation:

2^6,3^2,5^2|a^2=f,d^2=e,b^a=bf,g^a=g^2,g^c=g^2,h^a=g^2h,h^c=h^2,i^a=i^4,i^d=i^3,i^e=i^4,j^a=i^2j,j^d=j^3,j^e=j^4

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

holomorph of

Dic₁₅, C₂×D₁₅

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 3, 5, 5]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> SetConjugate(coll, 9, 4, F.9^3);;
gap> SetConjugate(coll, 9, 5, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9^2*F.10);;
gap> SetConjugate(coll, 10, 4, F.10^3);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);