C₅×#11771 #11934491

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11771

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1505
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₂³×C₅	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#405 ◁ G

292 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 94 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)(8,19,22,48)(9,15)(10,30,32,46)(11,17)(12,35)(13,18)(14,21)(16,33,44,34)(20,47,50,65)(23,43,24,54)(25,55,51,69)(26,57,58,67)(27,41)(28,61)(29,42)(31,45)(36,49)(37,52)(38,53)(39,56)(40,59,66,60)(62,64)(63,68);(6,8,13,22)(7,14,18,38)(9,25,29,51)(10,31,32,52)(11,24,12,23)(15,39,42,64)(16,43,44,54)(17,45,35,37)(19,46,48,30)(20,28,50,27)(21,33,53,34)(26,49,58,68)(36,61,63,41)(40,65,66,47)(55,67,69,57)(56,59,62,60);(6,9,12,28,13,29,11,27)(7,15,35,61,18,42,17,41)(8,20,24,51,22,50,23,25)(10,26,34,60,32,58,33,59)(14,36,45,64,38,63,37,39)(16,40,46,67,44,66,30,57)(19,47,54,69,48,65,43,55)(21,49,31,62,53,68,52,56)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4759);