C₅×#7195 #11929915

order	1280
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#7195

Subgroups

	subgroup	G/H
center	C₄²×C₅	C₂²⋊C₄
commutator	C₂²	C₄×C₅×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅
socle	C₂²×C₅	#336
prime radical	C₂³×C₅	C₄×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

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

normal subgroups only

This group has 222 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) = 640

A minimal faithful permutation representation — 2 generators on 37 points:

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

Conjugacy classes & character table

440 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 183);