C₅×#1468 #10602976

order	640
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#1468

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₅
prime radical	C₂²×C₅×D₄	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

280 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 3 generators on 29 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11991625, #11991632, #11991894, …
maximal subgroup of	C₅×#8487, C₅×#8488, C₅×#8539, …
maximal normal subgroup of	C₅×#8487, C₅×#8488, C₅×#8539, …
maximal quotient of	C₅×#7654, C₅×#7655, C₅×#7658, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5058);