Finite Group Databasebeta

← Prev (#13099641) Next (#13099643) →

C₅²×#361 #13099642

order	1600
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅² · Schur multiplier (H₂) = C₂×C₅

Aliases

C₅²×#361

Subgroups

	subgroup	G/H
center, socle	C₂×C₅²	#102
commutator	C₈	C₂×C₄×C₅²
Frattini, cosocle	C₂×C₈	C₂²×C₅²
prime radical	C₂×C₅²×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅² ◁ C₂×C₄×C₅² ◁ C₂×C₅²×C₈ ◁ G

248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 400

A minimal faithful permutation representation — 2 generators on 26 points:

(1,2,3,4,5)(11,12,14,18,15,20,25,19)(13,21,23,17,24,16,22,26);(6,7,8,9,10)(12,16)(13,22)(14,25)(17,18)(19,26)(20,21)(23,24)

Conjugacy classes & character table

400 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₅⋊D₅×#361, #72253101, #361×#1043, …
maximal subgroup of	C₅⋊D₅×#361, C₅²×AGL(1,ℤ/16), #72253101
maximal normal subgroup of	C₅⋊D₅×#361, C₅²×AGL(1,ℤ/16), #72253101
maximal quotient of	C₅²×#1333

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 1994);