Dic₁₅₇₉₆ #72295699

order	63184
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₇₉₆

Splittings

Hall	C₁₁×C₃₅₉ ⋊ Q₁₆
Sylow	C₃₅₉ ⋊ Dic₄₄
Sylow	C₁₁ ⋊ Dic₁₄₃₆

Subgroups

	subgroup	G/H
center	C₂	D₁₅₇₉₆
commutator, cosocle	C₄×C₁₁×C₃₅₉	C₂²
Frattini	C₄	C₂×D₃₉₄₉
Fitting	C₈×C₁₁×C₃₅₉	C₂
socle, prime radical	C₂×C₁₁×C₃₅₉	D₄
hypercenter	C₈	D₃₉₄₉
hypocenter	C₁₁×C₃₅₉	Q₁₆
LCS	G ▷ C₄×C₁₁×C₃₅₉ ▷ C₂×C₁₁×C₃₅₉ ▷ C₁₁×C₃₅₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 386
transitive degree	μ_t(G) = 63184

A minimal faithful permutation representation — 2 generators on 386 points:

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

Polycyclic presentation:

2^4,11,359|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^10,f^a=f^358

Conjugacy classes & character table

15799 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, 2, 11, 359]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^358);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15796 #72295699