C₅×#1306 #10598357

order	640
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#1306

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#95
commutator	C₂³	C₄²×C₅
Frattini, cosocle	C₂³×C₄	C₂²×C₅
prime radical	C₂⁴×C₅	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

192 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 160

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11986785, #11986881, D₅×#1306, …
maximal subgroup of	C₅×#7395, C₅×#7427, C₂×C₅×#1306, …
maximal normal subgroup of	C₅×#7395, C₅×#7427, C₂×C₅×#1306, …
maximal quotient of	C₅×#7032, C₅×#7033, C₅×#7037, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 439);