C₅×#12118 #11934838

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12117
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12118

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1595
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₅×#2017	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ G

392 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 70 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)(8,20)(9,25)(10,29)(11,18)(12,17)(13,19)(14,22)(15,27)(16,32)(21,49)(23,57)(24,48)(26,60)(28,45)(30,38)(31,41)(33,46)(34,42)(35,44)(36,43)(37,64)(39,55)(40,56)(47,51)(50,58)(52,69)(53,67)(54,65)(59,63)(61,66)(62,68);(6,8,13,24)(7,14,19,40)(9,21,28,53)(10,30,33,55)(11,23,12,36)(15,37,42,66)(16,43,44,57)(17,39,18,38)(20,46,48,29)(22,35,56,32)(25,58,45,65)(26,61,31,64)(27,51,34,52)(41,67,60,49)(47,68,69,59)(50,62,54,63);(6,9)(7,15)(8,21)(10,31)(11,34)(12,27)(13,28)(14,37)(16,41)(17,45)(18,25)(19,42)(20,47)(22,54)(23,52)(24,53)(26,33)(29,59)(30,64)(32,62)(35,63)(36,51)(38,58)(39,65)(40,66)(43,67)(44,60)(46,68)(48,69)(49,57)(50,56)(55,61)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5106);