C₂×#347×C₁₃₁ #30662292

order	16768
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#347×C₁₃₁

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃₁	C₄²
commutator	C₄	C₂×C₄²×C₁₃₁
Frattini, cosocle	C₂×C₈	C₂³×C₁₃₁
socle	C₂²×C₁₃₁	#97
prime radical	C₂³×C₁₃₁	M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ G

168 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 149
transitive degree	μ_t(G) = 4192

A minimal faithful permutation representation — 3 generators on 149 points:

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

Conjugacy classes & character table

5764 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₃₁×#7266, C₁₃₁×#7267, C₂×#1355×C₁₃₁, …
maximal normal subgroup of	C₁₃₁×#7266, C₁₃₁×#7267, C₂×#1355×C₁₃₁, …
maximal quotient of	C₂×#1313×C₁₃₁, C₁₃₁×#11404, C₁₃₁×#11413, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16768, 841);