C₅×#7309 #11930029

order	1280
nilpotency class	4
isoclinism	isoclinic class of #7307
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#7309

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#342
commutator	C₂²×C₄	C₄²×C₅
Frattini, cosocle	C₂²×#41	C₂²×C₅
prime radical	C₂²×C₅×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×#41 ◁ G

470 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 78 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)(9,11)(10,13)(14,15)(16,24,26,35)(17,28,30,21)(18,22)(19,23)(20,25)(27,39,42,45)(29,34)(31,38,40,36)(32,37)(33,41)(43,44);(6,8)(7,10,11,12)(9,13)(14,16,18,27,19,26,32,42)(15,20,22,33,23,34,37,44)(17,29,31,43,30,25,40,41)(21,35,36,45,28,24,38,39)

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, 297);