C₆₁×#11835 #29515268

order	15616
nilpotency class	4
isoclinism	isoclinic class of #11799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#11835

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₆₁
prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄×Q₈×C₆₁ ◁ G

340 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 101
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 3 generators on 101 points:

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

Conjugacy classes & character table

1708 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 4823);