C₅×#7495 #11930215

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7493
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7495

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1390
commutator	C₂×D₄	C₄²×C₅
Frattini, cosocle	C₂×#124	C₂²×C₅
prime radical	C₂²×C₅×D₄	C₂×C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#124 ◁ G

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

normal subgroups only

This group has 280 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)(6,7,9,14,11,15,23,34)(8,17,19,35,18,30,36,32)(10,16,26,33,25,13,22,31)(12,27,28,20,29,21,37,24);(6,8)(7,12,15,29)(9,21)(10,24)(11,18)(13,30,16,17)(14,32,34,35)(19,26)(20,25)(22,36)(23,27)(28,31,37,33)

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, 483);