#72242829

order	2560
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#1913 ⋊ C₄
Sylow	C₅ ⋊ #146614

Subgroups

	subgroup	G/H
center	C₂	#13021741
commutator	C₂×C₅×D₄	C₂×C₄²
Frattini	C₂²×D₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#1913	C₄
socle	C₂×C₅	#10745
prime radical	D₅×#534	C₂²
cosocle	C₂²×D₄×D₅	C₂³
hypercenter	#1913	AGL(1,𝔽₅)
hypocenter	C₅	#146614
LCS	G ▷ C₂×C₅×D₄ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ #1913

2442 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:

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

Polycyclic presentation:

2^9,5|a^2=i,b^2=d,c^2=e,g^2=i,b^a=bh,c^a=cf,c^b=cfgi,e^a=ei,f^b=fh,f^c=fi,g^a=gh,g^b=gh,g^c=gi,g^f=gi,h^c=hi,j^b=j^3,j^d=j^4

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#12753, #72242920, #72243005
maximal normal subgroup of	AGL(1,𝔽₅)×#12753, #72242920, #72243005

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