#72269860

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#1781 ⋊ C₄
Sylow	C₅ ⋊ #355096

Subgroups

	subgroup	G/H
center	C₂	#13027943
commutator	C₂³×C₅	C₂²×C₄²
Frattini	C₂⁴	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#1781	C₄
socle	C₂×C₅	#13775
prime radical	C₂³×D₄×D₅	C₂²
cosocle	C₂⁴×D₅	C₂⁴
hypercenter	#1781	AGL(1,𝔽₅)
hypocenter	C₅	#355096
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #1781

5520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 21 points:

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

Polycyclic presentation:

2^9,5|a^2=e,b^2=ef,c^a=ch,c^b=ci,d^a=dg,d^b=dh,e^c=ei,f^c=fi,g^b=gi,h^a=hi,j^a=j^3,j^b=j^3,j^e=j^4

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72269861, AGL(1,𝔽₅)×#21776
maximal normal subgroup of	#72269861, AGL(1,𝔽₅)×#21776

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