C₅×#7313 #11930033

order	1280
nilpotency class	4
isoclinism	isoclinic class of #7313
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#7313

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1304
commutator	C₂²×C₄	C₄²×C₅
Frattini, cosocle	C₂²×#41	C₂²×C₅
prime radical	C₂²×C₅×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₂²×C₅×#41 ◁ G

470 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,3,4,5)(6,7,10,14)(8,16,18,33)(9,20,21,31)(11,15,24,32)(12,25,26,35)(13,23,29,22)(17,27,34,30)(19,28,36,37);(6,8)(7,12)(9,17)(10,18,24,36)(11,19)(13,28)(14,30,32,26)(15,27)(16,20)(21,34,23,25)(22,35)(29,33,31,37)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 301);