C₅×#3080 #10616675

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

Aliases

direct product

C₅×#3080

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×C₄²×C₅ ◁ G

384 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 200 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)(6,7,9,11)(8,12,13,10)(14,15,18,20)(16,24,26,29)(17,23,19,22)(21,27,28,25);(6,8,9,13)(7,10,11,12)(15,20)(21,28)(22,23)(24,29);(14,16)(15,21)(17,25)(18,26)(19,27)(20,28)(22,24)(23,29);(14,17,18,19)(15,22,20,23)(16,25,26,27)(21,24,28,29)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11996393, #11996400, #12140803, …
maximal subgroup of	C₅×#9351, C₅×#9352, C₅×#24724, …
maximal normal subgroup of	C₅×#9351, C₅×#9352, C₅×#24724, …
maximal quotient of	C₅×#20649, C₅×#20875, C₅×#21034, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18757);