C₅×#3202 #10616797

order	640
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#3202

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#41 ◁ G

378 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 45
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 45 points:

(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15,19,23)(16,20,25,34)(17,27,29,37)(18,30,32,22)(21,31,36,28)(24,38,39,45)(26,42,41,35)(33,43,44,40);(6,8)(7,10)(9,13)(11,12)(14,16,19,25)(15,20,23,34)(17,24,29,39)(18,26,32,41)(21,33,36,44)(22,35,30,42)(27,38,37,45)(28,40,31,43);(8,13)(10,12)(14,17,19,29)(15,21,23,36)(16,24,25,39)(18,31,32,28)(20,33,34,44)(22,37,30,27)(26,43,41,40)(35,45,42,38);(7,11)(10,12)(16,25)(20,34)(24,39)(26,41)(33,44)(35,42)(38,45)(40,43)

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12011436, #12011453, #12011487, …
maximal subgroup of	C₅×#12109, C₅×#12112, C₅×#12118, …
maximal normal subgroup of	C₅×#12109, C₅×#12112, C₅×#12118, …
maximal quotient of	C₅×#21771, C₅×#21947, C₅×#22054, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18879);