C₆₁×#12026 #29515459

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×#12026

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:

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

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, 5014);