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C₄×C₅²×M₁₆ #13103141

order	1600
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅² · Schur multiplier (H₂) = C₂×C₄×C₅

Aliases

C₄×C₅²×M₁₆

Subgroups

	subgroup	G/H
center	C₄²×C₅²	C₂²
commutator	C₂	C₂×C₄²×C₅²
Frattini, cosocle	C₂×C₄	C₂³×C₅²
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂³×C₅²	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅² ◁ G

568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 800

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2,3,4)(13,14,15,16,17);(5,6,7,8,9,10,11,12)(18,19,20,21,22);(6,10)(8,12)

Conjugacy classes & character table

1000 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72227631
maximal subgroup of	C₄×M₁₆×C₅⋊D₅, C₅²×#3151, #72220990, …
maximal normal subgroup of	C₄×M₁₆×C₅⋊D₅, C₅²×#3151, #72220990, …
maximal quotient of	C₅²×#1455

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 5493);