C₅×#2858 #10616453

order	640
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

direct product

C₅×#2858

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
prime radical	C₂×C₅×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

752 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 264 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) = 21
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 21 points:

(1,2,3,4,5)(7,8)(11,12)(16,19)(17,21);(6,7,9,8)(10,11,13,12)(14,15,18,20)(16,17,19,21);(10,11,13,12)(14,16,18,19)(15,17,20,21);(6,8,9,7)(10,12,13,11)(14,17)(15,19)(16,20)(18,21)

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11987656, #11987748, #11987840, …
maximal subgroup of	C₅×#7659, C₅×#7679, C₅×#7699, …
maximal normal subgroup of	C₅×#7659, C₅×#7679, C₅×#7699, …
maximal quotient of	C₅×#14687, C₅×#14700, C₅×#14708, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18535);