C₅×#1683 #10603191

order	640
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#1683

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₅
prime radical	C₂×C₅×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

160 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 320

A minimal faithful permutation representation — 3 generators on 29 points:

(1,2,3,4,5)(6,7)(8,10)(9,11)(12,13)(14,15,18,21)(16,22,25,29)(17,26,23,20)(19,27,28,24);(7,10)(11,12)(14,16,23,27,18,25,17,24)(15,19,26,29,21,28,20,22);(7,11)(9,13)(10,12)

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#1683, #12081614, #12081745, …
maximal subgroup of	C₂×C₅×#1683, C₅×#19110, C₅×#19126, …
maximal normal subgroup of	C₂×C₅×#1683, C₅×#19110, C₅×#19126, …
maximal quotient of	C₅×#8648, C₅×#9215, C₅×#9219, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5273);