C₅×#7503 #11930223

order	1280
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#7503

Subgroups

	subgroup	G/H
center	C₄×C₅	#354
commutator	C₄²	C₄²×C₅
Frattini, cosocle	#532	C₂²×C₅
socle	C₂×C₅	#1393
prime radical	C₅×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#532 ◁ G

270 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 66 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)(6,7,9,14)(8,17,19,34)(10,15,24,32)(11,16,22,33)(12,26,27,36)(13,23,30,25)(18,29,35,31)(20,28,21,37);(6,8)(7,12)(9,21,22,26)(10,20)(11,18)(13,29)(14,31,33,34)(15,17)(16,28)(19,24,35,23)(25,36)(27,30,37,32)

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, 491);