#1814×C₁₃₁ #30661997

order	16768
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

#1814×C₁₃₁

Subgroups

	subgroup	G/H
center	C₄×C₁₃₁	#121
commutator	C₂×C₄	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₁₃₁
socle	C₂×C₁₃₁	#380
prime radical	C₂×#41×C₁₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ G

256 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 163
transitive degree	μ_t(G) = 4192

A minimal faithful permutation representation — 3 generators on 163 points:

(1,2)(3,12)(4,17,19,24)(5,20)(6,10)(7,11)(8,14)(9,18,25,21)(13,22,29,32)(15,23)(16,27)(26,28,31,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(1,3,5,14,7,16,6,15)(2,8,10,12,11,23,20,27)(4,13,18,28,19,29,21,30)(9,22,24,26,25,32,17,31);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,25)(12,26)(14,28)(15,30)(16,29)(17,20)(23,32)(27,31)

Conjugacy classes & character table

4192 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₃₁×#12747, C₁₃₁×#12750, C₁₃₁×#12751, …
maximal normal subgroup of	C₁₃₁×#12747, C₁₃₁×#12750, C₁₃₁×#12751, …
maximal quotient of	C₁₃₁×#8604, C₁₃₁×#8605, C₁₃₁×#8606, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16768, 546);