C₅×#12363 #11935083

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12363

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#446 ◁ G

278 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 1280

A minimal faithful permutation representation — 3 generators on 45 points:

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5351);