C₆₁×#12027 #29515460

order	15616
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#12027

Subgroups

	subgroup	G/H
center	C₄×C₆₁	#427
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₄×C₈	C₂³×C₆₁
socle	C₂×C₆₁	#1568
prime radical	C₂³×C₆₁	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ #97×C₆₁ ◁ C₆₁×#405 ◁ G

180 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 125
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 3 generators on 125 points:

(1,2,6,13,8,14,32,45,9,15,31,12,7,35,5,30)(3,16,19,37,21,47,18,38,22,43,33,61,20,55,53,10)(4,23,26,57,28,42,25,44,29,58,34,62,27,11,39,41)(17,48,50,36,52,63,60,64,24,59,54,40,51,46,49,56)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,3,5,18,7,20,31,19,9,22,32,53,8,21,6,33)(2,10,12,16,14,37,30,47,15,38,13,43,35,61,45,55)(4,24,25,60,27,52,26,50,29,17,39,49,28,51,34,54)(11,40,41,63,23,56,57,59,42,36,44,46,58,64,62,48);(1,4,7,27,9,29,8,28)(2,11,14,23,15,42,35,58)(3,17,20,51,22,24,21,52)(5,25,31,26,32,39,6,34)(10,36,37,64,38,40,61,56)(12,41,30,57,13,44,45,62)(16,46,47,48,43,63,55,59)(18,49,19,54,53,60,33,50)

Conjugacy classes & character table

3172 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 5015);