C₅²×#444 #13103181

order	1600
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅² · Schur multiplier (H₂) = C₂²×C₅

Aliases

direct product

C₅²×#444

Subgroups

	subgroup	G/H
center	C₄×C₅²	C₂×D₄
commutator	C₄	C₂²×C₄×C₅²
Frattini, cosocle	C₂×C₄	C₂³×C₅²
socle	C₂×C₅²	C₄×D₄
prime radical	C₅²×#135	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅² ◁ C₂×C₅²×C₈ ◁ G

416 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

550 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₅⋊D₅×#444, #72253093, #444×#1043, …
maximal subgroup of	C₅⋊D₅×#444, C₅²×AGL(1,ℤ/16), #72253093, …
maximal normal subgroup of	C₅⋊D₅×#444, C₅²×AGL(1,ℤ/16), #72253093, …
maximal quotient of	C₅²×#1580, C₂×C₅²×#444

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 5533);