C₅×#11868 #11934588

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11867
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#11868

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1519
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₅×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

396 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,7)(8,18)(9,23,26,12)(10,15)(11,17)(13,19)(14,24,28,36)(16,20)(21,30,25,29)(22,31)(27,37,33,34)(32,35);(7,12)(8,19)(9,24)(10,20)(13,29)(14,16)(15,28)(17,23)(18,33)(21,25)(22,35)(26,36)(27,31)(30,32);(6,8,20,35,11,22,10,19)(7,13,15,31,17,32,16,18)(9,25,36,34,26,21,24,37)(12,27,14,29,23,33,28,30)

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, 4856);