C₅×#12046 #11934766

order	1280
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12046

Subgroups

	subgroup	G/H
center	C₅×C₁₆	D₈
commutator	C₈	C₂²×C₅×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂×C₅	C₈×D₈
prime radical	C₅×#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₁₆ ◁ C₂×C₅×C₁₆ ◁ C₄×C₅×C₁₆ ◁ G

216 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 106 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)(6,7,11,17,13,18,29,16,14,19,10,28,12,8,20,22)(9,15,24,31,26,32,37,30,27,33,23,36,25,21,34,35);(6,8)(7,13)(9,21)(10,16)(11,22)(12,19)(14,18)(15,26)(17,29)(20,28)(23,30)(24,35)(25,33)(27,32)(31,37)(34,36);(6,9,11,24,13,26,29,37,14,27,10,23,12,25,20,34)(7,15,17,31,18,32,16,30,19,33,28,36,8,21,22,35)

Conjugacy classes & character table

440 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5034);