#2214×C₁₈₁ #37530157

order	23168
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₈₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

#2214×C₁₈₁

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₈₁	#459
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₁₈₁
prime radical	#135×C₁₈₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₈₁ ◁ C₂×C₄×C₁₈₁ ◁ C₄×C₈×C₁₈₁ ◁ G

150 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 5792

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

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

Conjugacy classes & character table

4163 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23168, 946);