C₆₁×#7243 #29510676

order	15616
nilpotency class	4
isoclinism	isoclinic class of #7217
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#7243

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#340
commutator	C₂×C₈	C₄²×C₆₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₆₁
prime radical	C₂³×C₆₁	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

198 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 85
transitive degree	μ_t(G) = 7808

A minimal faithful permutation representation — 2 generators on 85 points:

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

Conjugacy classes & character table

3172 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 231);