C₆₁×#13052 #29516485

order	15616
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₆₁×#13052

Subgroups

	subgroup	G/H
center	C₄×C₆₁	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×#41	C₂³×C₆₁
socle	C₂×C₆₁	#1889
prime radical	C₆₁×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₂×#41×C₆₁ ◁ G

624 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 93 points:

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

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, 6040);