C₆₁×#7018 #29510451

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

Aliases

direct product

C₆₁×#7018

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	#95
commutator	C₂×C₄	C₄×C₈×C₆₁
Frattini, cosocle	C₄³	C₂²×C₆₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₄³×C₆₁ ◁ G

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

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 15616

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

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

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, 6);