#72213910

order	9600
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₅² ⋊ C₂×Dic₃
derived	C₅²×C₂²⋊A₄ ⋊ C₂×C₄
Sylow	C₅² ⋊ #73080

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₅²×C₂²⋊A₄	C₂×C₄
Frattini	C₂²	#72213888
Fitting	C₂⁴×C₅²	C₂×Dic₃
socle	C₂²×C₅²	C₂×SL(2,ℤ/4)
prime radical	#72213644	C₂
cosocle	C₂²⋊A₄×C₅⋊D₅	C₂²
LCS	G ▷ C₅²×C₂²⋊A₄
UCS	C₁

1160 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 38 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^b=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=gi,g^b=gh,g^d=h,h^a=hij,h^d=gh,i^b=j,i^d=ij,j^b=i,j^d=i

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214138
maximal normal subgroup of	#72214138

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, 2, 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.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 2, F.9);;
gap> SetConjugate(coll, 10, 4, F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);