Dic₁₃₆₂₁ #72289773

order	54484
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₆₂₁

Splittings

derived, Hall	C₅₃×C₂₅₇ ⋊ C₄
Sylow	C₂₅₇ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₂₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₆₂₁
commutator, hypocenter	C₅₃×C₂₅₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅₃×C₂₅₇	C₂
LCS	G ▷ C₅₃×C₂₅₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 314
transitive degree	μ_t(G) = 54484

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

Polycyclic presentation:

2^2,53,257|a^2=b,c^a=c^52,d^a=d^256

Conjugacy classes & character table

13624 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, 53, 257]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^52);;
gap> SetConjugate(coll, 4, 1, F.4^256);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13621 #72289773