C₆₁×#7017 #29510450

order	15616
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₆₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₆₁×#7017

Subgroups

	subgroup	G/H
center, socle	C₂³×C₆₁	#95
commutator	C₂×C₄	C₄×C₈×C₆₁
Frattini, cosocle	C₂²×C₄²	C₂²×C₆₁
prime radical	C₂⁴×C₆₁	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

522 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 3904

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

(1,2)(3,5)(4,7)(6,8)(9,10)(11,15,16,13)(12,14)(17,18,19,21,20,22,23,24)(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);(2,5,7,6)(9,11)(10,13)(12,16)(14,15)

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, 5);