C₅×#20691 #11943411

order	1280
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅×#20691

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂⁴	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂⁴
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅×Q₈ ◁ G

1266 conjugacy classes of subgroups; only the 446 normal subgroups are shown.

normal subgroups only

This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 446 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) = 45
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 45 points:

(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15)(16,20)(17,27,29,21)(18,22)(19,23)(24,38,40,32)(25,33)(26,34)(28,41,43,35)(30,36)(31,37)(39,45,42,44);(7,10)(11,13)(14,16,19,26)(15,20,23,34)(17,24,29,40)(18,30,31,25)(21,32,27,38)(22,36,37,33)(28,42,43,39)(35,45,41,44);(6,8,9,12)(7,11,10,13)(14,17)(15,21)(16,24)(18,28)(19,29)(20,32)(22,35)(23,27)(25,39)(26,40)(30,42)(31,43)(33,44)(34,38)(36,45)(37,41);(14,18,19,31)(15,22,23,37)(16,25,26,30)(17,28,29,43)(20,33,34,36)(21,35,27,41)(24,39,40,42)(32,44,38,45)

Conjugacy classes & character table

290 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 13679);