C₆₁×#10662 #29514095

order	15616
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#10662

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₁	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	#399
prime radical	C₂³×C₆₁	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

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

normal subgroups only

This group has 320 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) = 85
transitive degree	μ_t(G) = 7808

A minimal faithful permutation representation — 3 generators on 85 points:

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

Conjugacy classes & character table

2440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 3650);