C₅×#12808 #11935528

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#12808

Subgroups

	subgroup	G/H
center	C₄×C₅	#410
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×#41	C₂³×C₅
socle	C₂×C₅	#1799
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×#131 ◁ G

420 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

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

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, 5796);