C₅×Q₈×Q₁₆ #10616977

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

Aliases

direct product

C₅×Q₈×Q₁₆

Subgroups

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

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

normal subgroups only

This group has 336 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.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 4 generators on 29 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)(22,29,28,27,26,25,24,23);(14,22,18,26)(15,23,19,27)(16,24,20,28)(17,25,21,29)

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12012990, #12013018, #12013346, …
maximal subgroup of	C₅×#12354, C₅×#12358, C₅×#12406, …
maximal normal subgroup of	C₅×#12354, C₅×#12358, C₅×#12406, …
maximal quotient of	C₅×Q₈×#103, C₅×C₄⋊C₄×Q₁₆, C₅×Q₈×#107, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 19059);