Dic₄₃₃₂ #72215771

order	17328
automorphism	Aut(G) order 23704704
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₃₃₂

Splittings

Hall	C₃×C₃₆₁ ⋊ Q₁₆
Sylow	C₃₆₁ ⋊ Dic₁₂
Sylow	C₃ ⋊ Dic₁₄₄₄

Subgroups

	subgroup	G/H
center	C₂	D₄₃₃₂
commutator, cosocle	C₃×C₄×C₃₆₁	C₂²
Frattini	C₄×C₁₉	C₂×D₅₇
Fitting	C₃×C₈×C₃₆₁	C₂
socle, prime radical	C₂×C₃×C₁₉	D₇₆
hypercenter	C₈	D₁₀₈₃
hypocenter	C₃×C₃₆₁	Q₁₆
LCS	G ▷ C₃×C₄×C₃₆₁ ▷ C₂×C₃×C₃₆₁ ▷ C₃×C₃₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 380
transitive degree	μ_t(G) = 17328

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

Polycyclic presentation:

2^4,3,19^2|a^2=d,b^2=cd,c^2=d,f^19=g,b^a=bc,c^a=cd,e^a=e^2,f^a=f^18g^18,g^a=g^18

Conjugacy classes & character table

4335 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₆₆₄, Dic₁₂₉₉₆
maximal normal subgroup of	Dic₈₆₆₄
maximal quotient of	Dic₁₂₉₉₆

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 19, 19]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^18*F.7^18);;
gap> SetConjugate(coll, 7, 1, F.7^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4332 #72215771