D₂₉₅₆₄ #72293552

order	59128
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₉₅₆₄

Splittings

Fitting	C₄×C₁₉×C₃₈₉ ⋊ C₂
Hall	C₁₉×C₃₈₉ ⋊ D₄
Sylow	C₃₈₉ ⋊ D₇₆
Sylow	C₁₉ ⋊ D₁₅₅₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₃₉₁
commutator, socle, cosocle	C₂×C₁₉×C₃₈₉	C₂²
Fitting	C₄×C₁₉×C₃₈₉	C₂
hypercenter	C₄	D₇₃₉₁
hypocenter	C₁₉×C₃₈₉	D₄
LCS	G ▷ C₂×C₁₉×C₃₈₉ ▷ C₁₉×C₃₈₉
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 412
transitive degree	μ_t(G) = 29564

A minimal faithful permutation representation — 2 generators on 412 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412);(2,4)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(25,412)(26,411)(27,410)(28,409)(29,408)(30,407)(31,406)(32,405)(33,404)(34,403)(35,402)(36,401)(37,400)(38,399)(39,398)(40,397)(41,396)(42,395)(43,394)(44,393)(45,392)(46,391)(47,390)(48,389)(49,388)(50,387)(51,386)(52,385)(53,384)(54,383)(55,382)(56,381)(57,380)(58,379)(59,378)(60,377)(61,376)(62,375)(63,374)(64,373)(65,372)(66,371)(67,370)(68,369)(69,368)(70,367)(71,366)(72,365)(73,364)(74,363)(75,362)(76,361)(77,360)(78,359)(79,358)(80,357)(81,356)(82,355)(83,354)(84,353)(85,352)(86,351)(87,350)(88,349)(89,348)(90,347)(91,346)(92,345)(93,344)(94,343)(95,342)(96,341)(97,340)(98,339)(99,338)(100,337)(101,336)(102,335)(103,334)(104,333)(105,332)(106,331)(107,330)(108,329)(109,328)(110,327)(111,326)(112,325)(113,324)(114,323)(115,322)(116,321)(117,320)(118,319)(119,318)(120,317)(121,316)(122,315)(123,314)(124,313)(125,312)(126,311)(127,310)(128,309)(129,308)(130,307)(131,306)(132,305)(133,304)(134,303)(135,302)(136,301)(137,300)(138,299)(139,298)(140,297)(141,296)(142,295)(143,294)(144,293)(145,292)(146,291)(147,290)(148,289)(149,288)(150,287)(151,286)(152,285)(153,284)(154,283)(155,282)(156,281)(157,280)(158,279)(159,278)(160,277)(161,276)(162,275)(163,274)(164,273)(165,272)(166,271)(167,270)(168,269)(169,268)(170,267)(171,266)(172,265)(173,264)(174,263)(175,262)(176,261)(177,260)(178,259)(179,258)(180,257)(181,256)(182,255)(183,254)(184,253)(185,252)(186,251)(187,250)(188,249)(189,248)(190,247)(191,246)(192,245)(193,244)(194,243)(195,242)(196,241)(197,240)(198,239)(199,238)(200,237)(201,236)(202,235)(203,234)(204,233)(205,232)(206,231)(207,230)(208,229)(209,228)(210,227)(211,226)(212,225)(213,224)(214,223)(215,222)(216,221)(217,220)(218,219)

Polycyclic presentation:

2^3,19,389|b^2=c,b^a=bc,d^a=d^18,e^a=e^388

Conjugacy classes & character table

14785 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 19, 389]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^18);;
gap> SetConjugate(coll, 5, 1, F.5^388);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D29564 #72293552