C₂×C₁₀₃×#1658 #39924263

order	26368
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₁₀₃×#1658

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₀₃	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₃
prime radical	C₂×#127×C₁₀₃	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₃ ◁ C₂×C₄²×C₁₀₃ ◁ G

882 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 882 conjugacy classes of subgroups — too many to draw. Only its 238 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) = 137
transitive degree	μ_t(G) = 13184

A minimal faithful permutation representation — 4 generators on 137 points:

(1,2)(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);(4,9)(5,12)(6,7)(11,14)(13,15)(16,17)(20,24)(22,29)(25,33)(26,30)(27,31)(32,34);(3,4)(5,10)(6,14)(7,9)(8,11)(12,17)(13,18)(15,16)(19,20,23,26)(21,25,28,34)(22,30,29,24)(27,32,31,33);(3,5,8,13)(4,10,11,18)(6,12,7,15)(9,16,14,17)(19,21,23,28)(20,25,26,34)(22,31,29,27)(24,32,30,33)

Conjugacy classes & character table

4120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26368, 11976);