C₅×#3093 #10616688

order	640
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#3093

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₅×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

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

normal subgroups only

This group has 414 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) = 160

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12132114, #12132122, #12132130, …
maximal subgroup of	C₅×#24020, C₅×#24021, C₅×#24022, …
maximal normal subgroup of	C₅×#24020, C₅×#24021, C₅×#24022, …
maximal quotient of	C₅×#20701, C₅×#20845, C₅×#20910, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18770);