C₅×#1288 #10598339

order	640
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#1288

Subgroups

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

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

normal subgroups only

This group has 152 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)(8,10)(9,12)(11,13)(14,15)(16,22,23,19)(17,21)(18,20)(24,28,29,26)(25,27);(7,10,12,11)(14,16,17,24,18,23,25,29)(15,19,21,26,20,22,27,28)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#1288, #11989142, #11989146, …
maximal subgroup of	C₂×C₅×#1288, C₅×#7972, C₅×#7973, …
maximal normal subgroup of	C₂×C₅×#1288, C₅×#7972, C₅×#7973, …
maximal quotient of	C₅×#7019, C₅×#7023, C₅×#7024, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 421);