C₅×#2822 #10616417

order	640
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#2822

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

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

normal subgroups only

This group has 380 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) = 33
transitive degree	μ_t(G) = 640

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

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12000591, #12000618, #12000636, …
maximal subgroup of	C₅×#10080, C₅×#10084, C₅×#10087, …
maximal normal subgroup of	C₅×#10080, C₅×#10084, C₅×#10087, …
maximal quotient of	C₅×#14629, C₅×#14631, C₅×#14634, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18499);