C₄×Q₁₆×C₂₅ #13098588

order	1600
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₄×Q₁₆×C₂₅

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₅	D₄
commutator	C₄	C₂²×C₄×C₂₅
Frattini, cosocle	C₂×C₄×C₅	C₂³×C₅
socle, prime radical	C₂²×C₅	C₂×C₅×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₅ ◁ C₄²×C₂₅ ◁ G

165 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 165 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 45
transitive degree	μ_t(G) = 1600

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(5,6,7,8,9,10,11,12)(13,20,19,18,17,16,15,14);(5,13,9,17)(6,14,10,18)(7,15,11,19)(8,16,12,20)

Conjugacy classes & character table

700 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 940);