Dic₁₅₈₈₈ #72296059

order	63552
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₈₈₈

Splittings

Hall	C₃×C₃₃₁ ⋊ Q₆₄
Sylow	C₃₃₁ ⋊ Dic₄₈
Sylow	C₃ ⋊ Dic₅₂₉₆

Subgroups

	subgroup	G/H
center	C₂	D₁₅₈₈₈
commutator, cosocle	C₃×C₁₆×C₃₃₁	C₂²
Frattini	C₁₆	C₂×D₉₉₃
Fitting	C₃×C₃₂×C₃₃₁	C₂
socle, prime radical	C₂×C₃×C₃₃₁	D₁₆
hypercenter	C₃₂	D₉₉₃
hypocenter	C₃×C₃₃₁	Q₆₄
LCS	G ▷ C₃×C₁₆×C₃₃₁ ▷ C₃×C₈×C₃₃₁ ▷ C₃×C₄×C₃₃₁ ▷ C₂×C₃×C₃₃₁ ▷ C₃×C₃₃₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 398
transitive degree	μ_t(G) = 63552

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

Polycyclic presentation:

2^6,3,331|a^2=f,b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2,h^a=h^330

Conjugacy classes & character table

15891 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", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 331]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^330);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15888 #72296059