Dic₁₂₆₉₁ #72288063

order	50764
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₆₉₁

Splittings

derived, Hall	C₃₇×C₃₄₃ ⋊ C₄
Sylow	C₃₄₃ ⋊ Dic₃₇
Sylow	C₃₇ ⋊ Dic₃₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₂₆₉₁
commutator, hypocenter	C₃₇×C₃₄₃	C₄
Frattini	C₂×C₄₉	D₂₅₉
Fitting, cosocle	C₂×C₃₇×C₃₄₃	C₂
socle, prime radical	C₂×C₇×C₃₇	D₄₉
LCS	G ▷ C₃₇×C₃₄₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 384
transitive degree	μ_t(G) = 50764

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

Polycyclic presentation:

2^2,7,37,7^2|a^2=b,c^7=e,e^7=f,c^a=c^6e^6f^6,d^a=d^36,e^a=e^6f^6,f^a=f^6

Conjugacy classes & character table

12694 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, 7, 37, 7, 7]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^6*F.5^6*F.6^6);;
gap> SetConjugate(coll, 4, 1, F.4^36);;
gap> SetConjugate(coll, 5, 1, F.5^6*F.6^6);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12691 #72288063