#3261×C₁₈₁ #37531204

order	23168
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₈₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

#3261×C₁₈₁

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₈₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₈₁
prime radical	C₂³×C₁₈₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₈₁ ◁ #117×C₁₈₁ ◁ G

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

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 221
transitive degree	μ_t(G) = 11584

A minimal faithful permutation representation — 4 generators on 221 points:

(2,5)(3,7)(9,10,14,18)(11,15,20,29)(12,22,24,32)(13,25,27,17)(16,26,31,23)(19,33,34,40)(21,37,36,30)(28,38,39,35)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221);(1,2)(3,6)(4,5)(7,8)(9,11,14,20)(10,15,18,29)(12,19,24,34)(13,21,27,36)(16,28,31,39)(17,30,25,37)(22,33,32,40)(23,35,26,38);(2,5)(6,8)(9,12,14,24)(10,16,18,31)(11,19,20,34)(13,26,27,23)(15,28,29,39)(17,32,25,22)(21,38,36,35)(30,40,37,33);(1,3,4,7)(2,6,5,8)(11,20)(15,29)(19,34)(21,36)(28,39)(30,37)(33,40)(35,38)

Conjugacy classes & character table

4706 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23168, 1993);