C₆₁×#11915 #29515348

order	15616
nilpotency class	4
isoclinism	isoclinic class of #11915
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#11915

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#1533
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	#97	C₂³×C₆₁
prime radical	C₂×#127×C₆₁	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₄×M₁₆×C₆₁ ◁ G

326 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 77
transitive degree	μ_t(G) = 976

A minimal faithful permutation representation — 3 generators on 77 points:

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

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, 4903);