C₅×#11769 #11934489

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11769

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1505
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₅×#469	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#405 ◁ G

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

normal subgroups only

This group has 304 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)(8,13)(9,21,24,12)(10,15)(11,17)(14,22,26,35)(16,23)(18,28)(19,33,34,25)(20,30)(27,31,36,37)(29,32);(7,12)(8,18)(9,22)(10,23)(13,27)(14,16)(15,26)(17,21)(19,34)(20,32)(24,35)(25,28)(29,33)(30,36);(6,8,10,18,11,20,23,32)(7,13,15,28,17,30,16,29)(9,19,22,31,24,34,35,37)(12,25,14,27,21,33,26,36)

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, 4757);