C₆₁×#7248 #29510681

order	15616
nilpotency class	4
isoclinism	isoclinic class of #7213
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#7248

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#342
commutator	C₂²×C₄	C₄²×C₆₁
Frattini, cosocle	C₂×C₄×Q₈	C₂²×C₆₁
prime radical	C₂³×C₆₁	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄×Q₈×C₆₁ ◁ G

244 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 2 generators on 93 points:

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

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, 236);