C₆₁×#9008 #29512441

order	15616
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#9008

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₁	#120
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	#386
prime radical	C₆₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

398 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 118 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:

(1,2,7,10,6,3,5,9)(4,8,14,16,13,11,12,15)(17,18)(19,28,31,39)(20,32)(21,26)(22,36)(23,27)(24,30,40,35)(25,33)(29,44,47,38)(34,41)(37,42)(43,45,48,46)(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);(1,3,6,2)(4,11,13,8)(5,9,7,10)(12,15,14,16)(17,19,22,30,23,31,21,35)(18,24,36,28,27,40,26,39)(20,29,37,45,34,47,33,46)(25,38,32,43,42,44,41,48);(1,4,6,13)(2,8,3,11)(5,14,7,12)(9,16,10,15)(17,20,23,34)(18,25,27,42)(19,29,31,47)(21,33,22,37)(24,38,40,44)(26,41,36,32)(28,43,39,48)(30,45,35,46)

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, 1996);