C₆₁×#20691 #29524124

order	15616
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₆₁ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₆₁×#20691

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₆₁
Frattini, cosocle	C₂⁴	C₂⁴×C₆₁
prime radical	C₂⁴×C₆₁	C₂⁴
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×Q₈×C₆₁ ◁ G

1266 conjugacy classes of subgroups; only the 446 normal subgroups are shown.

normal subgroups only

This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 446 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 — 4 generators on 101 points:

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

Conjugacy classes & character table

3538 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 13679);