C₅×#11994 #11934714

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11994
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#11994

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1558
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 ◁ C₄×C₅×M₁₆ ◁ G

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

normal subgroups only

This group has 258 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) = 37
transitive degree	μ_t(G) = 160

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4982);