D₁₆₁₅₇ #44568203

order	32314
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₁₅₇
unambiguous semidirect product	C₁₀₇⋊D₁₅₁
unambiguous semidirect product	C₁₅₁⋊D₁₀₇

Splittings

Fitting, socle, derived, Hall	C₁₀₇×C₁₅₁ ⋊ C₂
Sylow	C₁₅₁ ⋊ D₁₀₇
Sylow	C₁₀₇ ⋊ D₁₅₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₀₇×C₁₅₁	C₂
LCS	G ▷ C₁₀₇×C₁₅₁
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 258
transitive degree	μ_t(G) = 16157

A minimal faithful permutation representation — 2 generators on 258 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);(2,107)(3,106)(4,105)(5,104)(6,103)(7,102)(8,101)(9,100)(10,99)(11,98)(12,97)(13,96)(14,95)(15,94)(16,93)(17,92)(18,91)(19,90)(20,89)(21,88)(22,87)(23,86)(24,85)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(109,258)(110,257)(111,256)(112,255)(113,254)(114,253)(115,252)(116,251)(117,250)(118,249)(119,248)(120,247)(121,246)(122,245)(123,244)(124,243)(125,242)(126,241)(127,240)(128,239)(129,238)(130,237)(131,236)(132,235)(133,234)(134,233)(135,232)(136,231)(137,230)(138,229)(139,228)(140,227)(141,226)(142,225)(143,224)(144,223)(145,222)(146,221)(147,220)(148,219)(149,218)(150,217)(151,216)(152,215)(153,214)(154,213)(155,212)(156,211)(157,210)(158,209)(159,208)(160,207)(161,206)(162,205)(163,204)(164,203)(165,202)(166,201)(167,200)(168,199)(169,198)(170,197)(171,196)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)(178,189)(179,188)(180,187)(181,186)(182,185)(183,184)

Polycyclic presentation:

2,107,151|b^a=b^106,c^a=c^150

Conjugacy classes & character table

8080 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₆₁₅₇
maximal normal subgroup of	C₂×D₁₆₁₅₇
maximal quotient of	Dic₁₆₁₅₇, C₂×D₁₆₁₅₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32314, 3);

D16157 #44568203