Dic₉₉₀₉ #72274030

order	39636
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₉₉₀₉

Splittings

derived, Hall	C₂₇×C₃₆₇ ⋊ C₄
Sylow	C₃₆₇ ⋊ Dic₂₇
Sylow	C₂₇ ⋊ Dic₃₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₉₉₀₉
commutator, hypocenter	C₂₇×C₃₆₇	C₄
Frattini	C₂×C₉	D₁₁₀₁
Fitting, cosocle	C₂×C₂₇×C₃₆₇	C₂
socle, prime radical	C₂×C₃×C₃₆₇	D₉
LCS	G ▷ C₂₇×C₃₆₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 398
transitive degree	μ_t(G) = 39636

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

Polycyclic presentation:

2^2,3,367,3^2|a^2=b,c^3=e,e^3=f,c^a=c^2e^2f^2,d^a=d^366,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

9912 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, 3, 367, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 4, 1, F.4^366);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic9909 #72274030