#72242909

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#1913 ⋊ C₄
Sylow	C₅ ⋊ #368072

Subgroups

	subgroup	G/H
center	C₂	#13028242
commutator	C₂⁴×C₅	C₂³×C₄
Frattini	C₂⁴	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#1913	C₄
socle	C₂×C₅	#14015
prime radical	D₅×#534	C₂²
cosocle	C₂⁴×D₅	C₂⁴
hypercenter	#1913	AGL(1,𝔽₅)
hypocenter	C₅	#368072
LCS	G ▷ C₂⁴×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #1913

3914 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)(6,8,10,16)(7,13,14,11)(9,18,19,15)(12,20,17,21);(6,9)(7,12,14,17)(8,15)(10,19)(11,20,13,21)(16,18);(2,3,4,5)(6,8)(7,11)(9,15)(10,16)(12,21)(13,14)(17,20)(18,19);(6,7)(8,13)(9,17)(10,14)(11,16)(12,19)(15,21)(18,20)

Polycyclic presentation:

2^9,5|b^2=e,c^2=fg,d^2=i,b^a=bi,c^a=cf,c^b=cfg,d^a=dh,d^b=di,d^c=di,f^b=fi,f^c=fi,f^d=fi,g^a=gi,g^b=gi,g^c=gi,g^d=gi,h^c=hi,j^b=j^3,j^e=j^4

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242920, AGL(1,𝔽₅)×#23895, #72269864, …
maximal normal subgroup of	#72242920, 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, 2, F.5);;
gap> SetPower(coll, 3, F.6*F.7);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*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, 2, F.10^3);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);