#72243039

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#2021 ⋊ C₄
Sylow	C₅ ⋊ #375775

Subgroups

	subgroup	G/H
center	C₂	#13028398
commutator	C₂⁴×C₅	C₂³×C₄
Frattini	C₂⁴	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#2021	C₄
socle	C₂×C₅	#14171
prime radical	D₅×#2021	C₂
cosocle	C₂⁴×D₅	C₂⁴
hypercenter	#2021	AGL(1,𝔽₅)
hypocenter	C₅	#375775
LCS	G ▷ C₂⁴×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #2021

4980 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:

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

Polycyclic presentation:

2^9,5|a^2=i,c^2=fi,d^2=e,b^a=bh,c^a=cg,c^b=cf,d^a=dhi,d^b=di,d^c=di,f^a=fi,g^c=gi,g^d=gi,h^c=hi,j^d=j^3,j^e=j^4

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#23895, #72269864
maximal normal subgroup of	AGL(1,𝔽₅)×#23895, #72269864

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