#2229×C₁₈₁ #37530172

order	23168
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₈₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

#2229×C₁₈₁

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₈₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₁₈₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₈₁ ◁ C₄²×C₁₈₁ ◁ C₄×C₈×C₁₈₁ ◁ G

124 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 23168

A minimal faithful permutation representation — 3 generators on 221 points:

(1,2,4,6)(3,7,8,5)(9,10,12,15)(11,19,20,37)(13,25,23,38)(14,28,24,18)(16,32,30,21)(17,34,31,27)(22,39,29,35)(26,36,40,33)(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);(1,3,4,8)(2,5,6,7)(10,15)(16,30)(17,31)(18,28)(19,37)(25,38)(33,36)(35,39);(9,11,12,20)(10,16,15,30)(13,26,23,40)(14,29,24,22)(17,35,31,39)(18,36,28,33)(19,38,37,25)(21,34,32,27)

Conjugacy classes & character table

5249 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23168, 961);