C₆₁×#33983 #29537416

order	15616
nilpotency class	5
isoclinism	isoclinic class of #33982
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₁×#33983

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	C₂²×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₆₁
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ #41×C₆₁ ◁ #131×C₆₁ ◁ C₆₁×#504 ◁ G

462 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 125
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 4 generators on 125 points:

(1,2,7,13)(3,9,14,30)(4,8,21,29)(5,22,24,51)(6,25,23,12)(10,34,36,18)(11,37,38,19)(15,26,40,52)(16,46,41,62)(17,47,42,33)(20,39,48,28)(27,50,35,49)(31,57,55,43)(32,59,56,44)(45,60,61,54)(53,64,58,63)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(2,8)(4,18)(5,19)(6,23)(9,26)(10,11)(12,39)(13,29)(15,43)(16,44)(17,42)(20,50)(21,34)(22,27)(24,37)(25,28)(30,52)(31,32)(33,60)(35,51)(36,38)(40,57)(41,59)(45,64)(46,53)(47,54)(48,49)(55,56)(58,62)(61,63);(1,3)(2,9)(4,15)(5,16)(6,17)(7,14)(8,26)(10,31)(11,32)(12,33)(13,30)(18,43)(19,44)(20,45)(21,40)(22,46)(23,42)(24,41)(25,47)(27,53)(28,54)(29,52)(34,57)(35,58)(36,55)(37,59)(38,56)(39,60)(48,61)(49,63)(50,64)(51,62);(3,14)(9,30)(15,40)(16,41)(17,42)(26,52)(31,55)(32,56)(33,47)(43,57)(44,59)(45,61)(46,62)(53,58)(54,60)(63,64)

Conjugacy classes & character table

3172 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 26971);