#72229732

order	3840
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₅×AGL(1,ℤ/8) ⋊ C₂×C₄
Hall	C₃×C₅ ⋊ C₄×#453
Sylow	C₅ ⋊ #10801064
Sylow	C₃ ⋊ AGL(1,𝔽₅)×#453

Subgroups

	subgroup	G/H
center	C₂	#14321405
commutator	C₂×C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×AGL(1,ℤ/8)	C₂×C₄
socle	C₂×C₃×C₅	C₄×#120
prime radical	D₅×#5198	C₂
cosocle	C₂×C₃×C₄×D₅	C₂⁴
hypercenter	AGL(1,ℤ/8)	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	C₄×#453
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

3156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 4 generators on 16 points:

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

Polycyclic presentation:

2^8,3,5|d^2=e,g^2=h,b^a=bg,c^a=ch,c^b=cf,d^a=dh,d^b=df,f^a=fh,g^a=gh,g^b=gh,g^c=gh,g^d=gh,i^c=i^2,j^d=j^3,j^e=j^4

Conjugacy classes & character table

123 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×AGL(1,𝔽₅)×#453
maximal normal subgroup of	D₃×AGL(1,𝔽₅)×#453

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