C₆₁×#13666 #29517099

order	15616
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#13666

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₆₁
prime radical	C₂×C₆₁×#508	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂²×C₈×C₆₁ ◁ C₂²×C₁₆×C₆₁ ◁ G

288 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 129
transitive degree	μ_t(G) = 7808

A minimal faithful permutation representation — 3 generators on 129 points:

(1,2)(3,4)(5,6,11,17)(7,18,23,49)(8,13,26,16)(9,27,30,61)(10,31,33,39)(12,34,37,20)(14,40,43,29)(15,44,46,25)(19,48,22,55)(21,53,54,67)(24,56,59,60)(28,45,63,68)(32,41,51,66)(35,52,47,50)(36,57,64,62)(38,65,42,58)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(5,7)(6,12)(8,19)(9,28)(10,32)(11,23)(13,35)(14,41)(15,45)(16,47)(17,37)(18,27)(20,29)(21,25)(22,26)(24,57)(30,63)(31,36)(33,51)(34,40)(38,53)(39,64)(42,67)(43,66)(44,54)(46,68)(48,56)(49,61)(50,58)(52,65)(55,60)(59,62);(2,4)(5,8,11,26)(6,13,17,16)(7,19,23,22)(9,24,30,59)(10,25,33,44)(12,35,37,47)(14,38,43,42)(15,39,46,31)(18,48,49,55)(20,50,34,52)(21,51,54,32)(27,56,61,60)(28,57,63,62)(29,58,40,65)(36,45,64,68)(41,53,66,67)

Conjugacy classes & character table

2989 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 6654);