C₅×Q₈×SD₁₆ #10616974

order	640
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

C₅×Q₈×SD₁₆

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₂×C₅×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×Q₈ ◁ G

370 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.

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

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 19056);