C₆₁×#13606 #29517039

order	15616
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#13606

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#2127
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	#124	C₂³×C₆₁
prime radical	C₆₁×#3221	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#536 ◁ G

320 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 1952

A minimal faithful permutation representation — 3 generators on 93 points:

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

Conjugacy classes & character table

1525 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 6594);