C₆₁×#11557 #29514990

order	15616
nilpotency class	4
isoclinism	isoclinic class of #11541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₆₁×#11557

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#99
commutator	C₂⁴	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₆₁
prime radical	C₆₁×#2408	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄×D₄×C₆₁ ◁ G

750 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 1952

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

(1,2,4,8,5,6,3,7)(9,10)(11,15,16,13)(12,14)(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)(3,4)(10,13)(14,15);(1,3,5,4)(2,7,6,8)

Conjugacy classes & character table

2257 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 4545);