D₃₀₂₅₁ #72295567

order	60502
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₀₂₅₁

Splittings

Fitting, derived, Hall	C₁₆₉×C₁₇₉ ⋊ C₂
Sylow	C₁₇₉ ⋊ D₁₆₉
Sylow	C₁₆₉ ⋊ D₁₇₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₆₉×C₁₇₉	C₂
Frattini	C₁₃	D₂₃₂₇
socle	C₁₃×C₁₇₉	D₁₃
LCS	G ▷ C₁₆₉×C₁₇₉
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 348
transitive degree	μ_t(G) = 30251

A minimal faithful permutation representation — 2 generators on 348 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)(170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348);(2,169)(3,168)(4,167)(5,166)(6,165)(7,164)(8,163)(9,162)(10,161)(11,160)(12,159)(13,158)(14,157)(15,156)(16,155)(17,154)(18,153)(19,152)(20,151)(21,150)(22,149)(23,148)(24,147)(25,146)(26,145)(27,144)(28,143)(29,142)(30,141)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,134)(38,133)(39,132)(40,131)(41,130)(42,129)(43,128)(44,127)(45,126)(46,125)(47,124)(48,123)(49,122)(50,121)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)(171,348)(172,347)(173,346)(174,345)(175,344)(176,343)(177,342)(178,341)(179,340)(180,339)(181,338)(182,337)(183,336)(184,335)(185,334)(186,333)(187,332)(188,331)(189,330)(190,329)(191,328)(192,327)(193,326)(194,325)(195,324)(196,323)(197,322)(198,321)(199,320)(200,319)(201,318)(202,317)(203,316)(204,315)(205,314)(206,313)(207,312)(208,311)(209,310)(210,309)(211,308)(212,307)(213,306)(214,305)(215,304)(216,303)(217,302)(218,301)(219,300)(220,299)(221,298)(222,297)(223,296)(224,295)(225,294)(226,293)(227,292)(228,291)(229,290)(230,289)(231,288)(232,287)(233,286)(234,285)(235,284)(236,283)(237,282)(238,281)(239,280)(240,279)(241,278)(242,277)(243,276)(244,275)(245,274)(246,273)(247,272)(248,271)(249,270)(250,269)(251,268)(252,267)(253,266)(254,265)(255,264)(256,263)(257,262)(258,261)(259,260)

Polycyclic presentation:

2,13,179,13|b^13=d,b^a=b^12d^12,c^a=c^178,d^a=d^12

Conjugacy classes & character table

15127 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, 13, 179, 13]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2^12*F.4^12);;
gap> SetConjugate(coll, 3, 1, F.3^178);;
gap> SetConjugate(coll, 4, 1, F.4^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30251 #72295567