C₆₁×#7433 #29510866

order	15616
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#7433

Subgroups

	subgroup	G/H
center	C₄×C₆₁	#353
commutator	C₂×M₁₆	C₂×C₄×C₆₁
Frattini, cosocle	#407	C₂²×C₆₁
socle	C₂×C₆₁	#1352
prime radical	C₂⁴×C₆₁	Q₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#407 ◁ G

182 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 1952

A minimal faithful permutation representation — 2 generators on 93 points:

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

Conjugacy classes & character table

1891 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 421);