Dic₄₇₃₄ #72216183

order	18936
automorphism	Aut(G) order 29767392
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₇₃₄

Splittings

Hall	C₉×C₂₆₃ ⋊ Q₈
Sylow	C₂₆₃ ⋊ Dic₁₈
Sylow	C₉ ⋊ Dic₅₂₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₃₆₇
commutator, cosocle	C₂×C₉×C₂₆₃	C₂²
Frattini	C₂×C₃	C₂×D₇₈₉
Fitting	C₄×C₉×C₂₆₃	C₂
socle, prime radical	C₂×C₃×C₂₆₃	C₂×D₃
hypercenter	C₄	D₂₃₆₇
hypocenter	C₉×C₂₆₃	Q₈
LCS	G ▷ C₂×C₉×C₂₆₃ ▷ C₉×C₂₆₃
UCS	C₁ ◁ C₂ ◁ C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 280
transitive degree	μ_t(G) = 18936

A minimal faithful permutation representation — 2 generators on 280 points:

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,17)(11,16)(12,15)(13,14)(19,280)(20,279)(21,278)(22,277)(23,276)(24,275)(25,274)(26,273)(27,272)(28,271)(29,270)(30,269)(31,268)(32,267)(33,266)(34,265)(35,264)(36,263)(37,262)(38,261)(39,260)(40,259)(41,258)(42,257)(43,256)(44,255)(45,254)(46,253)(47,252)(48,251)(49,250)(50,249)(51,248)(52,247)(53,246)(54,245)(55,244)(56,243)(57,242)(58,241)(59,240)(60,239)(61,238)(62,237)(63,236)(64,235)(65,234)(66,233)(67,232)(68,231)(69,230)(70,229)(71,228)(72,227)(73,226)(74,225)(75,224)(76,223)(77,222)(78,221)(79,220)(80,219)(81,218)(82,217)(83,216)(84,215)(85,214)(86,213)(87,212)(88,211)(89,210)(90,209)(91,208)(92,207)(93,206)(94,205)(95,204)(96,203)(97,202)(98,201)(99,200)(100,199)(101,198)(102,197)(103,196)(104,195)(105,194)(106,193)(107,192)(108,191)(109,190)(110,189)(111,188)(112,187)(113,186)(114,185)(115,184)(116,183)(117,182)(118,181)(119,180)(120,179)(121,178)(122,177)(123,176)(124,175)(125,174)(126,173)(127,172)(128,171)(129,170)(130,169)(131,168)(132,167)(133,166)(134,165)(135,164)(136,163)(137,162)(138,161)(139,160)(140,159)(141,158)(142,157)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,150)

Polycyclic presentation:

2^3,3,263,3|a^2=c,b^2=c,d^3=f,b^a=bc,d^a=d^2f^2,e^a=e^262,f^a=f^2

Conjugacy classes & character table

4737 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₄₆₈, Dic₁₄₂₀₂
maximal normal subgroup of	Dic₉₄₆₈
maximal quotient of	Dic₁₄₂₀₂

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, 3, 263, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2);;
gap> SetConjugate(coll, 5, 1, F.5^262);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4734 #72216183