C₅×#12840 #11935560

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12840

Subgroups

	subgroup	G/H
center	C₄×C₅	#427
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1808
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ C₂×C₅×#131 ◁ G

354 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 154 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 — 3 generators on 69 points:

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

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5828);