#1535×C₁₈₁ #37529478

order	23168
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₈₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

#1535×C₁₈₁

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₈₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₁₈₁
Frattini, cosocle	C₄²	C₂³×C₁₈₁
prime radical	#121×C₁₈₁	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₈₁ ◁ C₂×C₄²×C₁₈₁ ◁ G

200 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 213
transitive degree	μ_t(G) = 11584

A minimal faithful permutation representation — 3 generators on 213 points:

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

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, 267);