Dic₄₅₁₈ #72215993

order	18072
automorphism	Aut(G) order 27108000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₅₁₈

Splittings

Hall	C₉×C₂₅₁ ⋊ Q₈
Sylow	C₂₅₁ ⋊ Dic₁₈
Sylow	C₉ ⋊ Dic₅₀₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₂₅₉
commutator, cosocle	C₂×C₉×C₂₅₁	C₂²
Frattini	C₂×C₃	C₂×D₇₅₃
Fitting	C₄×C₉×C₂₅₁	C₂
socle, prime radical	C₂×C₃×C₂₅₁	C₂×D₃
hypercenter	C₄	D₂₂₅₉
hypocenter	C₉×C₂₅₁	Q₈
LCS	G ▷ C₂×C₉×C₂₅₁ ▷ C₉×C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 268
transitive degree	μ_t(G) = 18072

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

Polycyclic presentation:

2^3,3,251,3|a^2=c,b^2=c,d^3=f,b^a=bc,d^a=d^2f^2,e^a=e^250,f^a=f^2

Conjugacy classes & character table

4521 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₀₃₆, Dic₁₃₅₅₄
maximal normal subgroup of	Dic₉₀₃₆
maximal quotient of	Dic₁₃₅₅₄

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, 251, 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.5^250);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4518 #72215993