#72214138

order	19200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₅²×C₂²⋊A₄	C₂²×C₄
Frattini	C₂²	S₄×D₅⋊AGL(1,𝔽₅)
Fitting	C₂⁴×C₅²	C₂×C₄×D₃
socle	C₂²×C₅²	C₂×C₄×S₄
prime radical	#72213882	C₂
cosocle	C₂²⋊A₄×C₅⋊D₅	C₂³
LCS	G ▷ C₅²×C₂²⋊A₄
UCS	C₁

4302 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 4302 conjugacy classes of subgroups — too many to draw. Only its 71 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:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

holomorph of

C₂²×D₅

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 5, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 6, 3, F.6^3);;
gap> SetConjugate(coll, 6, 4, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 3, F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 10, 5, F.11);;
gap> SetConjugate(coll, 11, 2, F.10*F.11);;
gap> SetConjugate(coll, 11, 5, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);