C₅×#7185 #11929905

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7185

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#95
commutator	C₂³	C₄×C₅×C₈
Frattini, cosocle	C₂³×C₈	C₂²×C₅
socle	C₂²×C₅	#336
prime radical	C₂⁴×C₅	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

314 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 2 generators on 45 points:

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

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, 173);