D₂₇₅₄₇ #72290874

order	55094
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₇₅₄₇

Splittings

Fitting, derived, Hall	C₁₆₃×C₁₆₉ ⋊ C₂
Sylow	C₁₆₉ ⋊ D₁₆₃
Sylow	C₁₆₃ ⋊ D₁₆₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₆₃×C₁₆₉	C₂
Frattini	C₁₃	D₂₁₁₉
socle	C₁₃×C₁₆₃	D₁₃
LCS	G ▷ C₁₆₃×C₁₆₉
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 332
transitive degree	μ_t(G) = 27547

A minimal faithful permutation representation — 2 generators on 332 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);(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)(171,332)(172,331)(173,330)(174,329)(175,328)(176,327)(177,326)(178,325)(179,324)(180,323)(181,322)(182,321)(183,320)(184,319)(185,318)(186,317)(187,316)(188,315)(189,314)(190,313)(191,312)(192,311)(193,310)(194,309)(195,308)(196,307)(197,306)(198,305)(199,304)(200,303)(201,302)(202,301)(203,300)(204,299)(205,298)(206,297)(207,296)(208,295)(209,294)(210,293)(211,292)(212,291)(213,290)(214,289)(215,288)(216,287)(217,286)(218,285)(219,284)(220,283)(221,282)(222,281)(223,280)(224,279)(225,278)(226,277)(227,276)(228,275)(229,274)(230,273)(231,272)(232,271)(233,270)(234,269)(235,268)(236,267)(237,266)(238,265)(239,264)(240,263)(241,262)(242,261)(243,260)(244,259)(245,258)(246,257)(247,256)(248,255)(249,254)(250,253)(251,252)

Polycyclic presentation:

2,13,163,13|b^13=d,b^a=b^12d^12,c^a=c^162,d^a=d^12

Conjugacy classes & character table

13775 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, 163, 13]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2^12*F.4^12);;
gap> SetConjugate(coll, 3, 1, F.3^162);;
gap> SetConjugate(coll, 4, 1, F.4^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27547 #72290874