C₆₁×#8187 #29511620

order	15616
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₆₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₆₁×#8187

Subgroups

	subgroup	G/H
center	C₄²×C₆₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₆₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

428 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 7808

A minimal faithful permutation representation — 3 generators on 93 points:

(1,2,5,9,6,10,4,7)(3,8,12,15,13,16,11,14)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(2,7,10,9)(8,14,16,15)(17,19,20,25,21,26,27,31)(18,22,24,29,23,28,30,32);(1,3,5,12,6,13,4,11)(2,8,9,15,10,16,7,14)(18,23)(22,28)(24,30)(29,32)

Conjugacy classes & character table

5368 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 1175);