C₅×#11391 #11934111

order	1280
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#11391

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#120
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂²×C₅	#379
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ G

842 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 254 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)(14,15,18,23,19,22,31,36)(16,24,27,37,26,20,30,33)(17,21,29,34,28,35,41,45)(25,32,40,43,39,38,42,44);(6,7,9,11)(8,12,13,10)(14,16)(15,20)(17,25)(18,30)(19,26)(21,32)(22,24)(23,37)(27,31)(28,39)(29,42)(33,36)(34,44)(35,38)(40,41)(43,45);(6,8,9,13)(7,10,11,12)(14,17)(15,21)(16,25)(18,29)(19,28)(20,32)(22,35)(23,34)(24,38)(26,39)(27,40)(30,42)(31,41)(33,43)(36,45)(37,44)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4379);