C₆₁×#8791 #29512224

order	15616
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₆₁×#8791

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#382
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₂³×C₆₁	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

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

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 3904

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

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

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, 1779);