C₆₁×#33981 #29537414

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

Aliases

direct product

C₆₁×#33981

Subgroups

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

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

normal subgroups only

This group has 466 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)(3,14)(4,10)(5,23)(6,27)(7,9)(8,16)(11,25)(12,21)(13,22)(15,42)(17,47)(18,49)(19,41)(20,50)(24,39)(26,55)(28,37)(29,36)(30,46)(31,43)(32,57)(33,59)(34,54)(35,52)(38,63)(40,45)(44,61)(48,64)(51,60)(53,56)(58,62)(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);(1,3)(2,8)(4,15)(5,24)(6,28)(7,19)(9,30)(10,31)(11,37)(12,39)(13,40)(14,23)(16,25)(17,21)(18,22)(20,51)(26,58)(27,32)(29,44)(33,61)(34,62)(35,47)(36,60)(38,64)(41,55)(42,50)(43,52)(45,63)(46,59)(48,54)(49,56)(53,57);(2,9)(8,30)(10,13)(11,33)(12,34)(14,41)(23,55)(27,36)(31,40)(32,60)(35,38)(37,61)(39,62)(42,49)(47,64)(50,56);(1,4,7,22)(2,10,9,13)(3,15,19,18)(5,20,26,53)(6,21,29,54)(8,31,30,40)(11,35,33,38)(12,36,34,27)(14,42,41,49)(16,43,46,45)(17,44,48,28)(23,50,55,56)(24,51,58,57)(25,52,59,63)(32,39,60,62)(37,47,61,64)

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, 26969);