C₅×#11913 #11934633

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11912
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#11913

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1530
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	#97	C₂³×C₅
prime radical	C₅×#3176	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#405 ◁ G

296 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 94 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)(6,7,9,15,10,12,24,26,11,16,23,32,25,14,21,31)(8,17,19,34,20,29,36,33,18,30,22,37,35,13,27,28);(7,12)(9,21)(10,25)(13,17)(14,16)(15,32)(19,27)(20,35)(22,36)(23,24)(28,33)(29,30);(6,8,11,18)(7,13)(9,22,23,19)(10,20,25,35)(12,17)(14,30)(15,33)(16,29)(21,36,24,27)(26,37)(28,32)(31,34)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4901);