C₆₁×#7047 #29510480

order	15616
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#7047

Subgroups

	subgroup	G/H
center, socle	C₂³×C₆₁	#95
commutator	C₂²×C₄	C₄²×C₆₁
Frattini, cosocle	C₂⁴×C₄	C₂²×C₆₁
prime radical	C₂⁵×C₆₁	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂⁴×C₄×C₆₁ ◁ G

608 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 2 generators on 93 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,15,16,13)(12,14)(17,18,20,25,22,26,21,24)(19,27,28,23,30,31,29,32)(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)(2,5,6,7)(4,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,29,22,30,20,28)(18,23,25,31,26,32,24,27)

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, 35);