Dic₃₆₄₈ #72214780

order	14592
automorphism	Aut(G) order 16809984
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₆₄₈

Splittings

Hall	C₃×C₁₉ ⋊ Q₂₅₆
Sylow	C₁₉ ⋊ Dic₁₉₂
Sylow	C₃ ⋊ Dic₁₂₁₆

Subgroups

	subgroup	G/H
center	C₂	D₃₆₄₈
commutator, cosocle	C₃×C₁₉×C₆₄	C₂²
Frattini	C₆₄	C₂×D₅₇
Fitting	C₃×C₁₉×C₁₂₈	C₂
socle, prime radical	C₂×C₃×C₁₉	D₆₄
hypercenter	C₁₂₈	D₅₇
hypocenter	C₃×C₁₉	Q₂₅₆
LCS	G ▷ C₃×C₁₉×C₆₄ ▷ C₃×C₁₉×C₃₂ ▷ C₃×C₁₆×C₁₉ ▷ C₃×C₈×C₁₉ ▷ C₃×C₄×C₁₉ ▷ C₂×C₃×C₁₉ ▷ C₃×C₁₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄ ◁ C₁₂₈

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 278
transitive degree	μ_t(G) = 14592

A minimal faithful permutation representation — 2 generators on 278 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)(129,256,255,254,253,252,251,250,249,248,247,246,245,244,243,242,241,240,239,238,237,236,235,234,233,232,231,230,229,228,227,226,225,224,223,222,221,220,219,218,217,216,215,214,213,212,211,210,209,208,207,206,205,204,203,202,201,200,199,198,197,196,195,194,193,192,191,190,189,188,187,186,185,184,183,182,181,180,179,178,177,176,175,174,173,172,171,170,169,168,167,166,165,164,163,162,161,160,159,158,157,156,155,154,153,152,151,150,149,148,147,146,145,144,143,142,141,140,139,138,137,136,135,134,133,132,131,130)(257,258,259)(260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278);(1,129,65,193)(2,130,66,194)(3,131,67,195)(4,132,68,196)(5,133,69,197)(6,134,70,198)(7,135,71,199)(8,136,72,200)(9,137,73,201)(10,138,74,202)(11,139,75,203)(12,140,76,204)(13,141,77,205)(14,142,78,206)(15,143,79,207)(16,144,80,208)(17,145,81,209)(18,146,82,210)(19,147,83,211)(20,148,84,212)(21,149,85,213)(22,150,86,214)(23,151,87,215)(24,152,88,216)(25,153,89,217)(26,154,90,218)(27,155,91,219)(28,156,92,220)(29,157,93,221)(30,158,94,222)(31,159,95,223)(32,160,96,224)(33,161,97,225)(34,162,98,226)(35,163,99,227)(36,164,100,228)(37,165,101,229)(38,166,102,230)(39,167,103,231)(40,168,104,232)(41,169,105,233)(42,170,106,234)(43,171,107,235)(44,172,108,236)(45,173,109,237)(46,174,110,238)(47,175,111,239)(48,176,112,240)(49,177,113,241)(50,178,114,242)(51,179,115,243)(52,180,116,244)(53,181,117,245)(54,182,118,246)(55,183,119,247)(56,184,120,248)(57,185,121,249)(58,186,122,250)(59,187,123,251)(60,188,124,252)(61,189,125,253)(62,190,126,254)(63,191,127,255)(64,192,128,256)(258,259)(261,278)(262,277)(263,276)(264,275)(265,274)(266,273)(267,272)(268,271)(269,270)

Polycyclic presentation:

2^8,3,19|a^2=h,b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,i^a=i^2,j^a=j^18

Conjugacy classes & character table

3651 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₂₉₆, Dic₁₀₉₄₄
maximal normal subgroup of	Dic₇₂₉₆
maximal quotient of	Dic₁₀₉₄₄

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", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 3, 19]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7*F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.10^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3648 #72214780