C₆₁×#12072 #29515505

order	15616
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#12072

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₄×C₈	C₂³×C₆₁
socle, prime radical	C₂²×C₆₁	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₄×C₈×C₆₁ ◁ C₈²×C₆₁ ◁ G

210 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 15616

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

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

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, 5060);