C₅×#2642 #10616237

order	640
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₅×#2642

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

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

normal subgroups only

This group has 416 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) = 33
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 4 generators on 33 points:

(1,2,3,4,5)(6,7,8,9)(12,16)(14,17)(19,23)(24,26)(25,31)(32,33);(6,7,8,9)(10,11,13,15)(12,14,16,17)(18,19)(20,26)(21,25)(22,23)(24,28)(27,33)(29,32)(30,31);(6,7)(8,9)(10,12)(11,14)(13,16)(15,17)(18,20,22,28)(19,24,23,26)(21,29,30,27)(25,33,31,32);(18,21,22,30)(19,25,23,31)(20,27,28,29)(24,32,26,33)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12000691, #12000747, #12074523, …
maximal subgroup of	C₅×#10095, C₅×#10103, C₅×#18441, …
maximal normal subgroup of	C₅×#10095, C₅×#10103, C₅×#18441, …
maximal quotient of	C₅×#14261, C₅×#14274, C₅×#14283, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18319);