D₁₅₈₅₇ #44540508

order	31714
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₈₅₇
unambiguous semidirect product	C₁₀₁⋊D₁₅₇
unambiguous semidirect product	C₁₅₇⋊D₁₀₁

Splittings

Fitting, socle, derived, Hall	C₁₀₁×C₁₅₇ ⋊ C₂
Sylow	C₁₅₇ ⋊ D₁₀₁
Sylow	C₁₀₁ ⋊ D₁₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₀₁×C₁₅₇	C₂
LCS	G ▷ C₁₀₁×C₁₅₇
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 258
transitive degree	μ_t(G) = 15857

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);(2,101)(3,100)(4,99)(5,98)(6,97)(7,96)(8,95)(9,94)(10,93)(11,92)(12,91)(13,90)(14,89)(15,88)(16,87)(17,86)(18,85)(19,84)(20,83)(21,82)(22,81)(23,80)(24,79)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(103,258)(104,257)(105,256)(106,255)(107,254)(108,253)(109,252)(110,251)(111,250)(112,249)(113,248)(114,247)(115,246)(116,245)(117,244)(118,243)(119,242)(120,241)(121,240)(122,239)(123,238)(124,237)(125,236)(126,235)(127,234)(128,233)(129,232)(130,231)(131,230)(132,229)(133,228)(134,227)(135,226)(136,225)(137,224)(138,223)(139,222)(140,221)(141,220)(142,219)(143,218)(144,217)(145,216)(146,215)(147,214)(148,213)(149,212)(150,211)(151,210)(152,209)(153,208)(154,207)(155,206)(156,205)(157,204)(158,203)(159,202)(160,201)(161,200)(162,199)(163,198)(164,197)(165,196)(166,195)(167,194)(168,193)(169,192)(170,191)(171,190)(172,189)(173,188)(174,187)(175,186)(176,185)(177,184)(178,183)(179,182)(180,181)

Polycyclic presentation:

2,101,157|b^a=b^100,c^a=c^156

Conjugacy classes & character table

7930 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₈₅₇
maximal normal subgroup of	C₂×D₁₅₈₅₇
maximal quotient of	Dic₁₅₈₅₇, C₂×D₁₅₈₅₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31714, 3);

D15857 #44540508