C₆₁×#7070 #29510503

order	15616
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#7070

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₁	#102
commutator	C₈	C₂×C₁₆×C₆₁
Frattini, cosocle	C₈²	C₂²×C₆₁
socle	C₂²×C₆₁	D₄⋊C₈
prime radical	C₂×D₈×C₆₁	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₄×C₈×C₆₁ ◁ C₈²×C₆₁ ◁ G

156 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Conjugacy classes & character table

3904 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 58);