C₅×#12207 #11934927

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12093
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12207

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂×C₅×C₂²⋊C₄ ◁ G

388 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 74 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)(6,7,9,12)(8,10,13,11)(14,15)(16,24)(17,21)(18,22)(19,23)(20,26)(25,41)(27,40)(28,39)(29,35)(30,36)(31,38)(32,42)(33,43)(34,44)(37,45);(7,10)(8,13)(11,12)(14,16,19,27)(15,20,23,33)(17,28,30,25)(18,26,31,43)(21,34,36,32)(22,24,38,40)(29,44,45,42)(35,39,37,41);(14,17,19,30)(15,21,23,36)(16,25,27,28)(18,29,31,45)(20,32,33,34)(22,37,38,35)(24,39,40,41)(26,42,43,44)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5195);