C₅×#7438 #11930158

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7438

Subgroups

	subgroup	G/H
center	C₄×C₅	#353
commutator	C₂×M₁₆	C₂×C₄×C₅
Frattini, cosocle	#408	C₂²×C₅
socle	C₂×C₅	#1352
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#408 ◁ G

174 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 42 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)(6,7,10,15)(8,18,20,44)(9,14,24,37)(11,17,29,42)(12,16,28,43)(13,31,32,52)(19,33,47,40)(21,45,50,67)(22,35,27,56)(23,34,51,60)(25,39,54,41)(26,38,53,65)(30,36,58,64)(46,62,57,69)(48,61,49,66)(55,63,59,68);(6,8)(7,13)(9,23,25,52)(10,27,28,57)(11,22)(12,21)(14,35,39,62)(15,40,43,66)(16,34)(17,33)(18,38,45,63)(19,26,48,55)(20,29,50,58)(24,47)(30,46)(31,59)(32,42,60,64)(36,61)(37,44)(41,67)(49,54)(51,53)(56,65)(68,69)

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, 426);