C₆₁×#8126 #29511559

order	15616
nilpotency class	3
isoclinism	isoclinic class of #8101
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#8126

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#95
commutator	C₂³	C₂×C₄²×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₂⁴×C₆₁	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

520 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 93
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 3 generators on 93 points:

(1,2,4,9,6,8,5,10)(3,11,12,16,13,7,14,15)(17,18,21,25,22,26,20,24)(19,27,30,31,29,23,28,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3,5,12,6,13,4,14)(2,7,10,15,8,11,9,16)(17,19)(18,23,26,27)(20,28)(21,30)(22,29)(24,31,25,32);(2,8)(7,11)(9,10)(15,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,27,32)

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, 1114);