C₆₁×#33982 #29537415

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

Aliases

direct product

C₆₁×#33982

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

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

normal subgroups only

This group has 472 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)(3,9)(4,8)(5,22)(6,25)(7,13)(10,18)(11,19)(12,23)(14,30)(15,26)(16,44)(17,45)(20,37)(21,29)(24,51)(27,47)(28,46)(31,41)(32,42)(33,40)(34,50)(35,48)(36,49)(38,52)(39,62)(43,58)(53,63)(54,61)(55,59)(56,60)(57,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(2,8)(4,18)(5,19)(6,23)(9,26)(10,11)(12,37)(13,29)(15,41)(16,42)(17,40)(20,50)(21,48)(22,27)(24,49)(25,28)(30,52)(31,32)(33,58)(34,51)(35,36)(38,59)(39,60)(43,64)(44,53)(45,54)(46,47)(55,56)(57,62)(61,63);(1,3)(2,9)(4,15)(5,16)(6,17)(7,14)(8,26)(10,31)(11,32)(12,33)(13,30)(18,41)(19,42)(20,43)(21,38)(22,44)(23,40)(24,39)(25,45)(27,53)(28,54)(29,52)(34,57)(35,55)(36,56)(37,58)(46,61)(47,63)(48,59)(49,60)(50,64)(51,62);(3,14)(9,30)(15,38)(16,39)(17,40)(26,52)(31,55)(32,56)(33,45)(41,59)(42,60)(43,61)(44,62)(53,57)(54,58)(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, 26970);