C₅×#2817 #10616412

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

Aliases

direct product

C₅×#2817

Subgroups

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

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

normal subgroups only

This group has 444 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) = 320

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

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11999944, #11999961, #12011358, …
maximal subgroup of	C₅×#9982, C₅×#9985, C₅×#12095, …
maximal normal subgroup of	C₅×#9982, C₅×#9985, C₅×#12095, …
maximal quotient of	C₅×#14626, C₅×#14627, C₅×#14629, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18494);