C₅×#11870 #11934590

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11867
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#11870

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

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

normal subgroups only

This group has 392 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)(6,7,10,16,12,17,11,27)(8,18,19,32,22,35,21,14)(9,13,23,15,26,30,25,29)(20,33,37,34,24,31,36,28);(7,13)(8,19)(9,23)(10,11)(14,31)(15,16)(17,30)(18,28)(21,22)(25,26)(27,29)(32,33)(34,35)(36,37);(6,8,11,21,12,22,10,19)(7,14,16,18,17,32,27,35)(9,24,23,36,26,20,25,37)(13,28,29,31,30,34,15,33)

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, 4858);