C₆₁×#12924 #29516357

order	15616
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#12924

Subgroups

	subgroup	G/H
center	C₄×C₆₁	#447
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂×C₆₁	#1852
prime radical	C₂⁴×C₆₁	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂²×M₁₆×C₆₁ ◁ G

386 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Conjugacy classes & character table

2440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 5912);