C₆₁×#7146 #29510579

order	15616
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#7146

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₁	#95
commutator	C₂×C₄	C₄×C₈×C₆₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₆₁
socle	C₂²×C₆₁	#336
prime radical	C₂³×C₆₁	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

210 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 7808

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

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

Conjugacy classes & character table

3904 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 134);