C₅×#3142 #10616737

order	640
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₅×#3142

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₅×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

392 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 320

A minimal faithful permutation representation — 4 generators on 29 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11996935, #12000181, #12000195, …
maximal subgroup of	C₅×#9442, C₅×#10019, C₅×#10021, …
maximal normal subgroup of	C₅×#9442, C₅×#10019, C₅×#10021, …
maximal quotient of	C₅×#20765, C₅×#21093, C₅×#21201, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18819);