C₅×#13436 #11936156

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13436
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#13436

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2044
commutator, Frattini, cosocle	C₂×#41	C₂³×C₅
prime radical	C₅×#3199	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

438 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 86 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)(9,13)(10,24)(11,15)(12,17)(14,21)(16,25)(19,32)(20,33)(22,34)(23,31)(26,28)(27,36)(29,35)(30,37);(7,13)(8,19)(9,11)(10,25)(15,17)(16,31)(20,22)(21,35)(24,28)(27,29)(32,33)(36,37);(6,8,9,20,12,22,11,19)(7,14,15,29,17,30,13,27)(10,21,23,36,25,37,26,35)(16,18,31,32,24,34,28,33)

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, 6424);