C₆₁×#7522 #29510955

order	15616
nilpotency class	6
isoclinism	isoclinic class of #7519
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#7522

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#1402
commutator	C₄⋊Q₈	C₂×C₄×C₆₁
Frattini, cosocle	#494	C₂²×C₆₁
prime radical	C₂²×C₆₁	AΣL(1,𝔽₁₆)
LCS	G ▷ C₄⋊Q₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂²×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₄²×C₆₁ ◁ C₆₁×#494 ◁ G

142 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 30 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:

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

Conjugacy classes & character table

1342 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 510);