C₆₁×#8799 #29512232

order	15616
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₆₁×#8799

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#382
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₂³×C₆₁	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

220 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 109
transitive degree	μ_t(G) = 7808

A minimal faithful permutation representation — 3 generators on 109 points:

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

Conjugacy classes & character table

2440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 1787);