C₆₁×#12808 #29516241

order	15616
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₆₁×#12808

Subgroups

	subgroup	G/H
center	C₄×C₆₁	#410
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×#41	C₂³×C₆₁
socle	C₂×C₆₁	#1799
prime radical	C₆₁×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂×C₈×C₆₁ ◁ C₂×#131×C₆₁ ◁ G

420 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 126 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)(3,12)(4,16)(5,10,19,8)(6,11)(7,14)(9,18)(13,26)(15,25,28,32)(17,23,30,24)(20,31,22,29)(21,27)(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);(1,3,4,14,6,13,18,27)(2,7,9,12,11,21,16,26)(5,15,17,29,19,28,30,31)(8,22,23,32,10,20,24,25);(2,8)(3,13)(5,19)(7,20)(9,23)(10,11)(12,25)(14,27)(16,24)(17,30)(21,22)(26,32)

Conjugacy classes & character table

2074 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 5796);