C₅×#3549 #10619334

order	640
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₅×#3549

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ G

1258 conjugacy classes of subgroups; only the 768 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 768 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) = 160

A minimal faithful permutation representation — 5 generators on 21 points:

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11991654, #11991855, #12015155, …
maximal subgroup of	C₅×#8491, C₅×#8530, C₅×#12769, …
maximal normal subgroup of	C₅×#8491, C₅×#8530, C₅×#12769, …
maximal quotient of	C₅×#34368, C₅×#34415, C₅×#34485, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 21416);