C₅×#12157 #11934877

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12155
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12157

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₅
prime radical	C₅×#537	C₂²
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂×C₅×C₄⋊C₄ ◁ G

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

normal subgroups only

This group has 372 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)(8,10)(9,12)(11,13)(14,15)(16,20)(17,27)(18,31)(19,23)(21,30)(22,32)(24,39)(25,38)(26,35)(28,43)(29,42)(33,40)(34,41)(36,44)(37,45);(7,10)(8,13)(11,12)(14,16,19,26)(15,20,23,35)(17,28,29,24)(18,25,32,41)(21,36,37,33)(22,38,31,34)(27,39,42,43)(30,44,45,40);(14,17)(15,21,23,37)(16,24)(18,30)(19,29)(20,33,35,36)(22,27,31,42)(25,40)(26,28)(32,45)(34,39,38,43)(41,44)

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, 5145);