C₆₁×#18728 #29522161

order	15616
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₁×#18728

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₆₁×#3548	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

920 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 7808

A minimal faithful permutation representation — 4 generators on 93 points:

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

Conjugacy classes & character table

2440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 11716);