C₅×#7423 #11930143

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×#7423

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	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×M₃₂ ◁ G

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

normal subgroups only

This group has 140 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)(8,19,21,50)(9,25,28,49)(10,32)(11,35)(12,38)(13,18)(14,20)(15,29,44,30)(16,27)(17,34)(22,55,23,56)(24,57,39,51)(26,46)(31,64,62,58)(33,67,36,43)(37,47)(40,65)(41,59)(42,54)(45,63,68,61)(48,53)(52,69)(60,66);(6,8,13,21)(7,14,18,42)(9,26,28,52)(10,33,11,36)(12,39,34,24)(15,43,44,67)(16,46,47,69)(17,48,38,41)(19,49,50,25)(20,29,54,30)(22,27,23,37)(31,65,62,60)(32,66,35,40)(45,56,68,55)(51,58,57,64)(53,63,59,61)

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, 411);