C₅×#11840 #11934560

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11753
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#11840

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₅
prime radical	C₅×#392	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#117 ◁ G

446 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 3 generators on 37 points:

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

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4828);