C₆₁×#13670 #29517103

order	15616
nilpotency class	5
isoclinism	isoclinic class of #13669
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#13670

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₆₁
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂×C₄×C₆₁ ◁ #131×C₆₁ ◁ C₆₁×#504 ◁ G

278 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 3 generators on 125 points:

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

Conjugacy classes & character table

2257 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 6658);