#4088

order	160
automorphism	Aut(G) = D₄²×AGL(1,𝔽₅) (order 1280) · Inn(G) = C₂×C₂²⋊D₅ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅×Q₈ ⋊ C₂
Sylow	C₅ ⋊ #137

Subgroups

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂³
Frattini	C₄	C₂²×D₅
Fitting	C₂×C₅×Q₈	C₂
socle	C₂×C₅	C₂×D₄
prime radical	#709	C₂
hypercenter	C₂×Q₈	D₅
hypocenter	C₅	#137
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

60 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:

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

Polycyclic presentation:

2^5,5|b^2=e,c^2=e,d^2=e,c^a=cd,d^a=de,f^a=f^4,c^b=ce,d^c=de

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#65043, #65044, #65350, …
maximal normal subgroup of	#65043, #65044, #65350, …
maximal quotient of	#65253, #65255, #65256, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 163);