D₁₆₁₆₇ #44568286

order	32334
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₁₆₇
unambiguous semidirect product	C₃⋊D₅₃₈₉
unambiguous semidirect product	C₁₇⋊D₉₅₁
unambiguous semidirect product	C₃₁₇⋊D₅₁

Splittings

Fitting, socle, derived, Hall	C₃×C₁₇×C₃₁₇ ⋊ C₂
Hall	C₁₇×C₃₁₇ ⋊ D₃
Hall	C₃×C₃₁₇ ⋊ D₁₇
Sylow	C₃₁₇ ⋊ D₅₁
Hall	C₃×C₁₇ ⋊ D₃₁₇
Sylow	C₁₇ ⋊ D₉₅₁
Sylow	C₃ ⋊ D₅₃₈₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₇×C₃₁₇	C₂
LCS	G ▷ C₃×C₁₇×C₃₁₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 337
transitive degree	μ_t(G) = 16167

A minimal faithful permutation representation — 2 generators on 337 points:

(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337);(2,3)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(22,337)(23,336)(24,335)(25,334)(26,333)(27,332)(28,331)(29,330)(30,329)(31,328)(32,327)(33,326)(34,325)(35,324)(36,323)(37,322)(38,321)(39,320)(40,319)(41,318)(42,317)(43,316)(44,315)(45,314)(46,313)(47,312)(48,311)(49,310)(50,309)(51,308)(52,307)(53,306)(54,305)(55,304)(56,303)(57,302)(58,301)(59,300)(60,299)(61,298)(62,297)(63,296)(64,295)(65,294)(66,293)(67,292)(68,291)(69,290)(70,289)(71,288)(72,287)(73,286)(74,285)(75,284)(76,283)(77,282)(78,281)(79,280)(80,279)(81,278)(82,277)(83,276)(84,275)(85,274)(86,273)(87,272)(88,271)(89,270)(90,269)(91,268)(92,267)(93,266)(94,265)(95,264)(96,263)(97,262)(98,261)(99,260)(100,259)(101,258)(102,257)(103,256)(104,255)(105,254)(106,253)(107,252)(108,251)(109,250)(110,249)(111,248)(112,247)(113,246)(114,245)(115,244)(116,243)(117,242)(118,241)(119,240)(120,239)(121,238)(122,237)(123,236)(124,235)(125,234)(126,233)(127,232)(128,231)(129,230)(130,229)(131,228)(132,227)(133,226)(134,225)(135,224)(136,223)(137,222)(138,221)(139,220)(140,219)(141,218)(142,217)(143,216)(144,215)(145,214)(146,213)(147,212)(148,211)(149,210)(150,209)(151,208)(152,207)(153,206)(154,205)(155,204)(156,203)(157,202)(158,201)(159,200)(160,199)(161,198)(162,197)(163,196)(164,195)(165,194)(166,193)(167,192)(168,191)(169,190)(170,189)(171,188)(172,187)(173,186)(174,185)(175,184)(176,183)(177,182)(178,181)(179,180)

Polycyclic presentation:

2,3,17,317|b^a=b^2,c^a=c^16,d^a=d^316

Conjugacy classes & character table

8085 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₆₁₆₇
maximal normal subgroup of	C₂×D₁₆₁₆₇
maximal quotient of	Dic₁₆₁₆₇, C₂×D₁₆₁₆₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32334, 8);

D16167 #44568286