D₅⋊C₃₂ #64841

order	320
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = AGL(1,𝔽₅)×#131 (order 640) · Inn(G) = AGL(1,𝔽₅) · Out(G) = #131
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

D₅⋊C₃₂

Splittings

derived, Sylow

C₅ ⋊ C₂×C₃₂

Subgroups

	subgroup	G/H
center, hypercenter	C₁₆	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂×C₃₂
Frattini	C₈	C₂×AGL(1,𝔽₅)
Fitting	C₅×C₁₆	C₄
socle	C₂×C₅	C₂×C₁₆
prime radical	C₂×D₅	C₁₆
cosocle	#672	C₂²
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₆

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,3,5,4)(6,8,10,19,11,20,23,33,12,9,18,21,24,22,32,34)(7,13,15,26,16,27,30,36,17,14,25,28,31,29,35,37);(2,3,4,5)(6,7,9,14,10,15,21,28,11,16,22,29,23,30,34,37,12,17,8,13,18,25,19,26,24,31,20,27,32,35,33,36)

Polycyclic presentation:

2^6,5|a^2=c,b^2=d,c^2=d,d^2=e,e^2=f,g^a=g^2,g^c=g^4

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×C₃₂, #10598486, #10598678, …
maximal normal subgroup of	AGL(1,𝔽₅)×C₃₂, #10598486, #10598678, …
maximal quotient of	D₅⋊C₆₄, #10598484, #10598652, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 179);

D5⋊C32 #64841