C₅×#7441 #11930161

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7440
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₅×#7441

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1353
commutator	C₂×M₁₆	C₂×C₄×C₅
Frattini, cosocle	#423	C₂²×C₅
prime radical	C₂³×C₅	#104
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#423 ◁ G

136 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 42 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)(8,16,18,28,20,33,19,36)(9,13,22,29)(10,24)(11,14)(12,17,26,34,27,35,32,21)(15,25)(23,31,37,30);(6,8)(7,12,14,27)(9,21,22,34)(10,19)(11,20)(13,28)(15,26,24,32)(16,31)(17,23,35,37)(18,25)(29,36)(30,33)

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, 429);