C₆₁×#7345 #29510778

order	15616
nilpotency class	5
isoclinism	isoclinic class of #1404
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#7345

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂×C₈×C₆₁
Frattini, cosocle	#391	C₂²×C₆₁
prime radical	C₂×2₋¹⁺⁴×C₆₁	C₄
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#391 ◁ G

342 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 85
transitive degree	μ_t(G) = 1952

A minimal faithful permutation representation — 2 generators on 85 points:

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

Conjugacy classes & character table

2074 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 333);