C₅×#7488 #11930208

order	1280
nilpotency class	5
isoclinism	isoclinic class of #1402
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7488

Subgroups

	subgroup	G/H
center	C₄×C₅	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₄²×C₅
Frattini, cosocle	#532	C₂²×C₅
socle	C₂×C₅	#1390
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#532 ◁ G

270 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 2 generators on 37 points:

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

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, 476);