C₅×#13368 #11936088

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13368

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

370 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 45 points:

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

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, 6356);