#72214037

order	9600
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂⁴×C₅² ⋊ C₄×D₃
derived	C₅²×C₂²⋊A₄ ⋊ C₂×C₄
Sylow	C₅² ⋊ C₄×#1020

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	C₅²×C₂²⋊A₄	C₂×C₄
Fitting, socle	C₂⁴×C₅²	C₄×D₃
prime radical	#72213721	C₂
cosocle	C₂²⋊A₄×C₅⋊D₅	C₂²
LCS	G ▷ C₅²×C₂²⋊A₄
UCS	C₁

2264 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 2264 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,5^2,2^4|b^2=c,d^a=d^2,e^a=e^4,e^b=e^3,e^c=e^4,f^a=e^3f,f^b=f^3,f^c=f^4,g^a=ghj,g^d=h,h^a=hi,h^d=gh,i^d=j,j^a=ij,j^d=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214138, #1020×D₅⋊AGL(1,𝔽₅)
maximal normal subgroup of	#72214138, #1020×D₅⋊AGL(1,𝔽₅)

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