C₅×#7252 #11929972

order	1280
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#7252

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#99
commutator	C₂²×C₄	C₄²×C₅
Frattini, cosocle	C₂×#117	C₂²×C₅
socle	C₂²×C₅	#342
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#117 ◁ G

344 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 102 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)(8,10)(9,12)(11,13)(14,15,18,22,19,23,31,38)(16,24,26,39,27,35,40,44)(17,21,29,36,30,37,42,45)(20,32,33,28,34,41,43,25);(7,10,12,11)(14,16)(15,20)(17,28,30,25)(18,26)(19,27)(21,35,37,24)(22,33)(23,34)(29,41,42,32)(31,40)(36,44,45,39)(38,43)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 240);