C₆₁×#13312 #29516745

order	15616
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₆₁×#13312

Subgroups

	subgroup	G/H
center	C₄×C₆₁	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂×C₆₁	#1999
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×#131×C₆₁ ◁ G

520 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 114 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)(4,21)(5,26)(6,13)(7,12)(8,14)(9,17)(10,24)(11,29)(16,45)(18,51)(19,39)(20,44)(22,54)(23,56)(25,37)(27,42)(28,41)(30,40)(31,38)(32,61)(33,49)(34,50)(35,46)(36,59)(43,47)(48,64)(52,55)(53,58)(57,62)(60,63)(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);(1,3)(2,9)(4,22)(5,27)(6,18)(7,19)(8,20)(10,35)(11,39)(12,33)(13,42)(14,34)(15,26)(16,24)(17,29)(21,52)(23,57)(25,47)(28,50)(30,49)(31,48)(32,37)(36,43)(38,62)(40,44)(41,51)(45,53)(46,59)(54,56)(55,58)(60,61)(63,64);(1,4)(2,10)(3,16)(5,23)(6,31)(7,24)(8,25)(9,32)(11,36)(12,37)(13,21)(14,38)(15,43)(17,46)(18,22)(19,47)(20,48)(26,53)(27,55)(28,60)(29,58)(30,59)(33,62)(34,52)(35,42)(39,64)(40,63)(41,56)(44,54)(45,51)(49,61)(50,57)

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, 6300);