C₅×#10636 #11933356

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10631
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#10636

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 278 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) = 320

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

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

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, 3624);