#72243003

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#1781 ⋊ C₄
Sylow	C₅ ⋊ #146479

Subgroups

	subgroup	G/H
center	C₂	#13021669
commutator	C₂²×C₄×C₅	C₂×C₄²
Frattini	C₂²×D₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#1781	C₄
socle	C₂×C₅	#10725
prime radical	C₂³×D₄×D₅	C₂²
cosocle	C₂²×D₄×D₅	C₂³
hypercenter	#1781	AGL(1,𝔽₅)
hypocenter	C₅	#146479
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ #1781

4952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

(1,2,5,3,4)(7,13)(8,17)(9,10)(12,20);(2,3,4,5)(6,8,9,19)(7,12,14,21)(10,18,11,17)(13,20,15,16);(2,3,4,5)(6,7)(8,16,17,21)(9,15)(10,14)(11,13)(12,19,20,18)

Polycyclic presentation:

2^9,5|a^2=d,b^2=de,g^2=i,b^a=bg,c^a=ci,c^b=cef,d^b=di,e^a=eh,e^b=ei,e^c=ei,f^a=fhi,f^c=fi,g^b=ghi,g^c=gi,g^e=gi,g^f=gi,h^b=hi,j^a=j^3,j^b=j^3,j^d=j^4

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243005, #72243033, AGL(1,𝔽₅)×#12707, …
maximal normal subgroup of	#72243005, #72243033, AGL(1,𝔽₅)×#12707, …

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