D₂₃₈₂₀ #72283554

order	47640
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₃₈₂₀

Splittings

Fitting	C₃×C₄×C₅×C₃₉₇ ⋊ C₂
Hall	C₃×C₅×C₃₉₇ ⋊ D₄
Hall	C₅×C₃₉₇ ⋊ D₁₂
Hall	C₃×C₃₉₇ ⋊ D₂₀
Sylow	C₃₉₇ ⋊ D₆₀
Hall	C₃×C₅ ⋊ D₁₅₈₈
Sylow	C₅ ⋊ D₄₇₆₄
Sylow	C₃ ⋊ D₇₉₄₀

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₉₅₅
commutator, socle, cosocle	C₂×C₃×C₅×C₃₉₇	C₂²
Fitting	C₃×C₄×C₅×C₃₉₇	C₂
hypercenter	C₄	D₅₉₅₅
hypocenter	C₃×C₅×C₃₉₇	D₄
LCS	G ▷ C₂×C₃×C₅×C₃₉₇ ▷ C₃×C₅×C₃₉₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 409
transitive degree	μ_t(G) = 23820

A minimal faithful permutation representation — 2 generators on 409 points:

(1,2,3,4)(5,6,7)(8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409);(2,4)(6,7)(9,12)(10,11)(14,409)(15,408)(16,407)(17,406)(18,405)(19,404)(20,403)(21,402)(22,401)(23,400)(24,399)(25,398)(26,397)(27,396)(28,395)(29,394)(30,393)(31,392)(32,391)(33,390)(34,389)(35,388)(36,387)(37,386)(38,385)(39,384)(40,383)(41,382)(42,381)(43,380)(44,379)(45,378)(46,377)(47,376)(48,375)(49,374)(50,373)(51,372)(52,371)(53,370)(54,369)(55,368)(56,367)(57,366)(58,365)(59,364)(60,363)(61,362)(62,361)(63,360)(64,359)(65,358)(66,357)(67,356)(68,355)(69,354)(70,353)(71,352)(72,351)(73,350)(74,349)(75,348)(76,347)(77,346)(78,345)(79,344)(80,343)(81,342)(82,341)(83,340)(84,339)(85,338)(86,337)(87,336)(88,335)(89,334)(90,333)(91,332)(92,331)(93,330)(94,329)(95,328)(96,327)(97,326)(98,325)(99,324)(100,323)(101,322)(102,321)(103,320)(104,319)(105,318)(106,317)(107,316)(108,315)(109,314)(110,313)(111,312)(112,311)(113,310)(114,309)(115,308)(116,307)(117,306)(118,305)(119,304)(120,303)(121,302)(122,301)(123,300)(124,299)(125,298)(126,297)(127,296)(128,295)(129,294)(130,293)(131,292)(132,291)(133,290)(134,289)(135,288)(136,287)(137,286)(138,285)(139,284)(140,283)(141,282)(142,281)(143,280)(144,279)(145,278)(146,277)(147,276)(148,275)(149,274)(150,273)(151,272)(152,271)(153,270)(154,269)(155,268)(156,267)(157,266)(158,265)(159,264)(160,263)(161,262)(162,261)(163,260)(164,259)(165,258)(166,257)(167,256)(168,255)(169,254)(170,253)(171,252)(172,251)(173,250)(174,249)(175,248)(176,247)(177,246)(178,245)(179,244)(180,243)(181,242)(182,241)(183,240)(184,239)(185,238)(186,237)(187,236)(188,235)(189,234)(190,233)(191,232)(192,231)(193,230)(194,229)(195,228)(196,227)(197,226)(198,225)(199,224)(200,223)(201,222)(202,221)(203,220)(204,219)(205,218)(206,217)(207,216)(208,215)(209,214)(210,213)(211,212)

Polycyclic presentation:

2^3,3,5,397|b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^396

Conjugacy classes & character table

11913 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", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 397]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^396);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D23820 #72283554