#1802×C₁₃₁ #30661985

order	16768
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

#1802×C₁₃₁

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₁₃₁
prime radical	C₂²×D₄×C₁₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 147
transitive degree	μ_t(G) = 8384

A minimal faithful permutation representation — 3 generators on 147 points:

(2,4)(5,6,7,8)(9,10)(11,13)(12,15)(14,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(1,2,3,4);(10,13)(11,16)(14,15)

Conjugacy classes & character table

4978 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₃₁×#8443, C₁₃₁×#8444, C₁₃₁×#8446, …
maximal normal subgroup of	C₁₃₁×#8443, C₁₃₁×#8444, C₁₃₁×#8446, …
maximal quotient of	C₁₃₁×#8563, C₁₃₁×#8567, C₁₃₁×#8571, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16768, 534);

#1802×C131 #30661985