C₅×#11990 #11934710

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11989
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#11990

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1551
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂³×C₅	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#405 ◁ G

268 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 98 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)(6,7,10,16,11,15,29,41,12,18,30,44,32,17,45,43)(8,19,21,47,22,36,51,61,23,35,52,42,31,13,28,34)(9,24,26,53,27,14,37,38,25,40,54,66,55,39,63,65)(20,33,49,59,50,46,67,68,48,60,62,69,57,58,56,64);(6,8,12,23)(7,13,18,36)(9,20,25,48)(10,28,30,51)(11,31,32,22)(14,33,39,60)(15,35,17,19)(16,42,44,47)(21,29,52,45)(24,46,40,58)(26,56,54,67)(27,57,55,50)(34,41,61,43)(37,62,63,49)(38,64,65,68)(53,59,66,69);(6,9)(7,14,18,39)(8,20)(10,26)(11,27)(12,25)(13,33,36,60)(15,40,17,24)(16,38,44,65)(19,46,35,58)(21,49)(22,50)(23,48)(28,56)(29,37)(30,54)(31,57)(32,55)(34,59,61,69)(41,66,43,53)(42,64,47,68)(45,63)(51,67)(52,62)

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, 4978);