C₆₁×#10678 #29514111

order	15616
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₆₁×#10678

Subgroups

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

478 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 154 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:

(1,2,4,6)(3,7,8,5)(9,10,12,14,13,16,11,15)(18,21)(22,23)(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);(1,3,4,8)(2,5,6,7)(10,14)(11,12)(15,16)(17,18,19,21)(20,22,24,23);(10,15)(11,12)(14,16)(18,22)(20,24)(21,23)

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, 3666);