C₅×#3131 #10616726

order	640
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₅×#3131

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂²×D₄
prime radical	C₅×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

400 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 192 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 — 4 generators on 29 points:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11994612, #11994627, #11998653, …
maximal subgroup of	C₅×#9067, C₅×#9070, C₅×#9751, …
maximal normal subgroup of	C₅×#9067, C₅×#9070, C₅×#9751, …
maximal quotient of	C₅×#20760, C₅×#21083, C₅×#21195, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18808);