C₅×#7425 #11930145

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7422
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7425

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1347
commutator	C₂×C₁₆	C₂×C₄×C₅
Frattini, cosocle	C₂×M₃₂	C₂²×C₅
prime radical	C₂³×C₅	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×M₃₂ ◁ G

136 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 69 points:

(1,2,3,4,5)(6,7,10,17,14,19,13,40)(8,20,22,54,26,55,25,61)(9,27,30,62,33,53,34,63)(11,36,35,52,12,38,29,18)(15,41,42,57,45,58,44,21)(16,46,48,31,50,65,49,32)(23,59,37,68,24,60,39,47)(28,51,64,67,56,69,66,43);(6,8,14,26)(7,15,19,45)(9,28,33,56)(10,25,13,22)(11,37,12,39)(16,47,50,68)(17,42,40,44)(18,51,52,69)(20,53,55,27)(21,31,57,32)(23,29,24,35)(30,66,34,64)(36,67,38,43)(41,46,58,65)(48,59,49,60)(54,63,61,62)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 413);