C₆₁×#12044 #29515477

order	15616
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#12044

Subgroups

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

222 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Conjugacy classes & character table

3904 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 5032);