C₆₁×#13668 #29517101

order	15616
nilpotency class	5
isoclinism	isoclinic class of #13667
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#13668

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₆₁
prime radical	C₆₁×#3414	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₂²×C₈×C₆₁ ◁ C₂×M₃₂×C₆₁ ◁ G

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

normal subgroups only

This group has 284 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:

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

Conjugacy classes & character table

1891 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 6656);