C₅×#9496 #11932216

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₅×#9496

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂⁵	C₂³×C₅
prime radical	C₂³×C₅×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂⁵×C₅ ◁ G

900 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 29
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 29 points:

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

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, 2484);