Dic₁₅₇₈₀ #72295656

order	63120
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₇₈₀

Splittings

Hall	C₃×C₅×C₂₆₃ ⋊ Q₁₆
Hall	C₅×C₂₆₃ ⋊ Dic₁₂
Hall	C₃×C₂₆₃ ⋊ Dic₂₀
Sylow	C₂₆₃ ⋊ Dic₆₀
Hall	C₃×C₅ ⋊ Dic₁₀₅₂
Sylow	C₅ ⋊ Dic₃₁₅₆
Sylow	C₃ ⋊ Dic₅₂₆₀

Subgroups

	subgroup	G/H
center	C₂	D₁₅₇₈₀
commutator, cosocle	C₃×C₄×C₅×C₂₆₃	C₂²
Frattini	C₄	C₂×D₃₉₄₅
Fitting	C₃×C₅×C₈×C₂₆₃	C₂
socle, prime radical	C₂×C₃×C₅×C₂₆₃	D₄
hypercenter	C₈	D₃₉₄₅
hypocenter	C₃×C₅×C₂₆₃	Q₁₆
LCS	G ▷ C₃×C₄×C₅×C₂₆₃ ▷ C₂×C₃×C₅×C₂₆₃ ▷ C₃×C₅×C₂₆₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 287
transitive degree	μ_t(G) = 63120

A minimal faithful permutation representation — 2 generators on 287 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,24)(22,23)(26,287)(27,286)(28,285)(29,284)(30,283)(31,282)(32,281)(33,280)(34,279)(35,278)(36,277)(37,276)(38,275)(39,274)(40,273)(41,272)(42,271)(43,270)(44,269)(45,268)(46,267)(47,266)(48,265)(49,264)(50,263)(51,262)(52,261)(53,260)(54,259)(55,258)(56,257)(57,256)(58,255)(59,254)(60,253)(61,252)(62,251)(63,250)(64,249)(65,248)(66,247)(67,246)(68,245)(69,244)(70,243)(71,242)(72,241)(73,240)(74,239)(75,238)(76,237)(77,236)(78,235)(79,234)(80,233)(81,232)(82,231)(83,230)(84,229)(85,228)(86,227)(87,226)(88,225)(89,224)(90,223)(91,222)(92,221)(93,220)(94,219)(95,218)(96,217)(97,216)(98,215)(99,214)(100,213)(101,212)(102,211)(103,210)(104,209)(105,208)(106,207)(107,206)(108,205)(109,204)(110,203)(111,202)(112,201)(113,200)(114,199)(115,198)(116,197)(117,196)(118,195)(119,194)(120,193)(121,192)(122,191)(123,190)(124,189)(125,188)(126,187)(127,186)(128,185)(129,184)(130,183)(131,182)(132,181)(133,180)(134,179)(135,178)(136,177)(137,176)(138,175)(139,174)(140,173)(141,172)(142,171)(143,170)(144,169)(145,168)(146,167)(147,166)(148,165)(149,164)(150,163)(151,162)(152,161)(153,160)(154,159)(155,158)(156,157)

Polycyclic presentation:

2^4,3,5,263|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4,g^a=g^262

Conjugacy classes & character table

15783 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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 263]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^262);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15780 #72295656