Dic₁₄₈₅₇ #72292168

order	59428
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₈₅₇

Splittings

derived, Hall	C₈₃×C₁₇₉ ⋊ C₄
Sylow	C₁₇₉ ⋊ Dic₈₃
Sylow	C₈₃ ⋊ Dic₁₇₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₈₅₇
commutator, hypocenter	C₈₃×C₁₇₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₈₃×C₁₇₉	C₂
LCS	G ▷ C₈₃×C₁₇₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 266
transitive degree	μ_t(G) = 59428

A minimal faithful permutation representation — 2 generators on 266 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);(1,3,2,4)(6,87)(7,86)(8,85)(9,84)(10,83)(11,82)(12,81)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(89,266)(90,265)(91,264)(92,263)(93,262)(94,261)(95,260)(96,259)(97,258)(98,257)(99,256)(100,255)(101,254)(102,253)(103,252)(104,251)(105,250)(106,249)(107,248)(108,247)(109,246)(110,245)(111,244)(112,243)(113,242)(114,241)(115,240)(116,239)(117,238)(118,237)(119,236)(120,235)(121,234)(122,233)(123,232)(124,231)(125,230)(126,229)(127,228)(128,227)(129,226)(130,225)(131,224)(132,223)(133,222)(134,221)(135,220)(136,219)(137,218)(138,217)(139,216)(140,215)(141,214)(142,213)(143,212)(144,211)(145,210)(146,209)(147,208)(148,207)(149,206)(150,205)(151,204)(152,203)(153,202)(154,201)(155,200)(156,199)(157,198)(158,197)(159,196)(160,195)(161,194)(162,193)(163,192)(164,191)(165,190)(166,189)(167,188)(168,187)(169,186)(170,185)(171,184)(172,183)(173,182)(174,181)(175,180)(176,179)(177,178)

Polycyclic presentation:

2^2,83,179|a^2=b,c^a=c^82,d^a=d^178

Conjugacy classes & character table

14860 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d");;
gap> coll := SingleCollector(F, [2, 2, 83, 179]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^82);;
gap> SetConjugate(coll, 4, 1, F.4^178);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14857 #72292168