C₆₁×#13489 #29516922

order	15616
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#13489

Subgroups

	subgroup	G/H
center	C₄×C₆₁	#482
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂×C₆₁	#2073
prime radical	C₂×#135×C₆₁	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#405 ◁ G

278 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

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, 6477);