C₅×#7449 #11930169

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7449

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1361
commutator	C₂×C₁₆	C₂×C₄×C₅
Frattini, cosocle	C₄×C₁₆	C₂²×C₅
prime radical	C₂²×C₅	#361
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

126 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 69 points:

(1,2,3,4,5)(6,7,10,16,33,19,35,45,13,20,34,18,12,36,11,17)(8,21,23,52,37,54,61,69,26,55,60,44,25,62,24,53)(9,27,29,15,43,46,56,48,32,64,67,49,31,47,30,63)(14,28,38,22,41,57,50,68,42,66,40,59,51,65,39,58);(6,8,13,26)(7,14,20,42)(9,28,32,66)(10,24,34,61)(11,23,35,60)(12,37,33,25)(15,44,49,52)(16,39,18,50)(17,38,45,40)(19,51,36,41)(21,46,55,47)(22,56,59,30)(27,62,64,54)(29,58,67,68)(31,57,43,65)(48,69,63,53)

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, 437);