C₅×#20340 #11943060

order	1280
nilpotency class	2
isoclinism	isoclinic class of #20338
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄²×C₅ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₅×#20340

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂²×C₄²
commutator	C₄	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂×C₅	C₂²×#95
prime radical	C₂²×C₅×#41	C₂²
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ G

1956 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.

normal subgroups only

This group has 1956 conjugacy classes of subgroups — too many to draw. Only its 1114 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 — 4 generators on 37 points:

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

Conjugacy classes & character table

410 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 13328);