C₆₁×#7356 #29510789

order	15616
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₆₁ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₆₁×#7356

Subgroups

	subgroup	G/H
center	C₄×C₆₁	#353
commutator	C₄²	C₂×C₈×C₆₁
Frattini, cosocle	#423	C₂²×C₆₁
socle	C₂×C₆₁	#1318
prime radical	C₆₁×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#423 ◁ G

202 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Conjugacy classes & character table

1708 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 344);