C₅×#1638 #10603146

order	640
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#1638

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₅
prime radical	C₂×C₅×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

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

normal subgroups only

This group has 194 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) = 37
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,7,11,14)(8,15,17,20)(9,13,10,18)(12,19,21,16)(22,23,27,31)(24,32,35,36)(25,30,26,29)(28,34,37,33);(6,8)(7,12)(9,16)(10,19)(11,17)(13,20)(14,21)(15,18)(22,24,27,35)(23,28,31,37)(25,34,26,33)(29,36,30,32);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)(23,29)(25,26)(28,36)(30,31)(32,37)(33,34)

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#1638, #12075071, #12075324, …
maximal subgroup of	C₂×C₅×#1638, C₅×#18494, C₅×#18521, …
maximal normal subgroup of	C₂×C₅×#1638, C₅×#18494, C₅×#18521, …
maximal quotient of	C₅×#7765, C₅×#7767, C₅×#7773, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5228);