Dic₁₂₆₆₀ #72288016

order	50640
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₆₆₀

Splittings

Hall	C₃×C₅×C₂₁₁ ⋊ Q₁₆
Hall	C₅×C₂₁₁ ⋊ Dic₁₂
Hall	C₃×C₂₁₁ ⋊ Dic₂₀
Sylow	C₂₁₁ ⋊ Dic₆₀
Hall	C₃×C₅ ⋊ Dic₈₄₄
Sylow	C₅ ⋊ Dic₂₅₃₂
Sylow	C₃ ⋊ Dic₄₂₂₀

Subgroups

	subgroup	G/H
center	C₂	D₁₂₆₆₀
commutator, cosocle	C₃×C₄×C₅×C₂₁₁	C₂²
Frattini	C₄	C₂×D₃₁₆₅
Fitting	C₃×C₅×C₈×C₂₁₁	C₂
socle, prime radical	C₂×C₃×C₅×C₂₁₁	D₄
hypercenter	C₈	D₃₁₆₅
hypocenter	C₃×C₅×C₂₁₁	Q₁₆
LCS	G ▷ C₃×C₄×C₅×C₂₁₁ ▷ C₂×C₃×C₅×C₂₁₁ ▷ C₃×C₅×C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 235
transitive degree	μ_t(G) = 50640

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

Polycyclic presentation:

2^4,3,5,211|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^210

Conjugacy classes & character table

12663 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, 211]);;
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.7^210);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12660 #72288016