C₆₁×#62694 #29566127

order	15616
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₆₁ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₆₁×#62694

Subgroups

	subgroup	G/H
center	C₄²×C₆₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₆₁
socle	C₂²×C₆₁	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₆₁ ◁ G

6640 conjugacy classes of subgroups; only the 5670 normal subgroups are shown.

normal subgroups only

This group has 6640 conjugacy classes of subgroups — too many to draw. Only its 5670 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) = 3904

A minimal faithful permutation representation — 6 generators on 93 points:

(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(3,10)(6,9)(11,16)(13,15)(18,23)(19,25)(24,30)(26,31);(2,7)(6,9)(8,14)(13,15)(19,25)(22,28)(26,31)(29,32);(17,20,21,27)(18,24,23,30)(19,26,25,31)(22,29,28,32);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)

Conjugacy classes & character table

4636 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 55682);