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C₄²×C₅ #691

order	80
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₄×GL(2,ℤ/4) (order 384) · Inn(G) = C₁ · Out(G) = C₄×GL(2,ℤ/4)
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₅)×AGL(2,ℤ/4)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₄

Aliases

C₄²×C₅

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂²×C₅
socle, prime radical	C₂²×C₅	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

center of	C₅×#322, C₄×C₅×M₁₆, C₅×#431, …
derived subgroup of	C₅⋊(#127⋊C₃), C₅⋊(#126⋊C₃), D₅×C₄²⋊C₃, …
Frattini subgroup of	C₈×#6014, #13097660, #13097661, …
Fitting subgroup of	C₄×#168, #3935, C₄⋊#168, …
maximal subgroup of	C₄×#168, #3935, C₄⋊#168, …
maximal normal subgroup of	C₄×#168, #3935, C₄⋊#168, …
maximal quotient of	C₅×#95, C₄×C₅×C₈, C₅×#97, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 20);