C₅×#7374 #11930094

order	1280
nilpotency class	5
isoclinism	isoclinic class of #1406
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₈ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#7374

Subgroups

	subgroup	G/H
center	C₄×C₅	#353
commutator	C₄²	C₂×C₅×C₈
Frattini, cosocle	#423	C₂²×C₅
socle	C₂×C₅	#1325
prime radical	C₂³×C₅	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#423 ◁ G

148 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 46 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)(6,7,9,15,11,16,25,32,12,17,24,33,10,14,26,30)(8,18,20,34,22,28,36,37,19,29,35,31,21,13,23,27);(6,8,10,21,12,19,11,22)(7,13,14,29,17,28,16,18)(9,23,25,20,24,36,26,35)(15,31,32,27,33,34,30,37)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 362);