C₅×#13580 #11936300

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13578
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#13580

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	C₂×#353
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	#124	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#532 ◁ G

402 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 90 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)(8,18,21,28)(9,14)(10,25)(11,16,27,32)(12,17)(13,20,29,37)(15,26,33,24)(19,34,36,30)(22,31,23,35);(6,8,12,23)(7,13,17,30)(9,20,10,19)(11,21,27,22)(14,31,25,28)(15,18,33,35)(16,34,32,29)(24,37,26,36);(6,9,12,10)(7,14,17,25)(8,19,23,20)(11,24,27,26)(13,28,30,31)(15,32,33,16)(18,34,35,29)(21,36,22,37)

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, 6568);