#1652×C₁₈₁ #37529595

order	23168
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₈₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

#1652×C₁₈₁

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₈₁	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₁₈₁
prime radical	C₂×D₄×C₁₈₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₈₁ ◁ C₄²×C₁₈₁ ◁ G

200 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 205
transitive degree	μ_t(G) = 11584

A minimal faithful permutation representation — 3 generators on 205 points:

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

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, 384);