C₅×#9196 #11931916

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9183
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#9196

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

478 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 134 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)(6,7,11,12,10,8,9,13)(14,15)(16,24)(17,28,30,21)(18,31)(19,23)(20,26)(22,33)(25,38,41,45)(27,39)(29,36,32,37)(34,40,44,42)(35,43);(6,8,10,7)(9,13,11,12)(14,16,18,26,19,27,33,43)(15,20,22,24,23,35,31,39)(17,25,32,42,30,41,29,40)(21,34,37,38,28,44,36,45);(6,9,10,11)(7,12,8,13)(14,17)(15,21)(16,25)(18,32)(19,30)(20,34)(22,37)(23,28)(24,38)(26,42)(27,41)(29,33)(31,36)(35,44)(39,45)(40,43)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2184);