C₆₁×#7505 #29510938

order	15616
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₆₁×#7505

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#354
commutator	C₄²	C₄²×C₆₁
Frattini, cosocle	C₂×C₄⋊Q₈	C₂²×C₆₁
prime radical	C₂³×C₆₁	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄⋊Q₈×C₆₁ ◁ G

260 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 97
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 2 generators on 97 points:

(5,6,8,13,10,15,22,32)(7,16,18,33,17,28,34,30)(9,14,23,31,24,12,21,29)(11,25,26,19,27,20,36,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(1,2,3,4)(5,7,9,19,10,17,24,35)(6,11,14,16,15,27,12,28)(8,20,21,18,22,25,23,34)(13,30,29,36,32,33,31,26)

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, 493);