C₆₁×#33728 #29537161

order	15616
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₆₁×#33728

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂³×C₆₁	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

482 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 270 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) = 7808

A minimal faithful permutation representation — 4 generators on 101 points:

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

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, 26716);