D₁₅₈₅₁ #44540199

order	31702
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₈₅₁
unambiguous semidirect product	C₁₂₁⋊D₁₃₁
unambiguous semidirect product	C₁₃₁⋊D₁₂₁

Splittings

Fitting, derived, Hall	C₁₂₁×C₁₃₁ ⋊ C₂
Sylow	C₁₃₁ ⋊ D₁₂₁
Sylow	C₁₂₁ ⋊ D₁₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₂₁×C₁₃₁	C₂
Frattini	C₁₁	D₁₄₄₁
socle	C₁₁×C₁₃₁	D₁₁
LCS	G ▷ C₁₂₁×C₁₃₁
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 252
transitive degree	μ_t(G) = 15851

A minimal faithful permutation representation — 2 generators on 252 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);(2,121)(3,120)(4,119)(5,118)(6,117)(7,116)(8,115)(9,114)(10,113)(11,112)(12,111)(13,110)(14,109)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,98)(26,97)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(123,252)(124,251)(125,250)(126,249)(127,248)(128,247)(129,246)(130,245)(131,244)(132,243)(133,242)(134,241)(135,240)(136,239)(137,238)(138,237)(139,236)(140,235)(141,234)(142,233)(143,232)(144,231)(145,230)(146,229)(147,228)(148,227)(149,226)(150,225)(151,224)(152,223)(153,222)(154,221)(155,220)(156,219)(157,218)(158,217)(159,216)(160,215)(161,214)(162,213)(163,212)(164,211)(165,210)(166,209)(167,208)(168,207)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)(175,200)(176,199)(177,198)(178,197)(179,196)(180,195)(181,194)(182,193)(183,192)(184,191)(185,190)(186,189)(187,188)

Polycyclic presentation:

2,11,131,11|b^11=d^10,b^a=b^10d,c^a=c^130,d^a=d^10

Conjugacy classes & character table

7927 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₈₅₁
maximal normal subgroup of	C₂×D₁₅₈₅₁
maximal quotient of	Dic₁₅₈₅₁, C₂×D₁₅₈₅₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31702, 4);

D15851 #44540199