C₅×#12729 #11935449

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12727
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#12729

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1782
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂²×Q₈	C₂³×C₅
prime radical	C₅×#535	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₅×#519 ◁ G

690 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 138 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)(6,7,10,15,11,16,23,31)(8,17,20,32,19,13,27,29)(9,12,22,26,21,14,24,30)(18,28,34,37,33,25,35,36);(7,12)(9,21)(10,23)(13,25)(14,16)(15,26)(17,28)(18,33)(20,27)(29,36)(30,31)(32,37);(6,8)(7,13,16,17)(9,18)(10,20)(11,19)(12,25,14,28)(15,29,31,32)(21,33)(22,34)(23,27)(24,35)(26,36,30,37)

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, 5717);