Dic₁₂₁₅₀ #72282005

order	48600
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₁₅₀

Splittings

Hall	C₂₅×C₂₄₃ ⋊ Q₈
Sylow	C₂₄₃ ⋊ Dic₅₀
Sylow	C₂₅ ⋊ Dic₄₈₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₆₀₇₅
commutator, cosocle	C₂×C₂₅×C₂₄₃	C₂²
Frattini	C₂×C₅×C₈₁	C₂×D₁₅
Fitting	C₄×C₂₅×C₂₄₃	C₂
socle, prime radical	C₂×C₃×C₅	C₂×D₄₀₅
hypercenter	C₄	D₆₀₇₅
hypocenter	C₂₅×C₂₄₃	Q₈
LCS	G ▷ C₂×C₂₅×C₂₄₃ ▷ C₂₅×C₂₄₃
UCS	C₁ ◁ C₂ ◁ C₄

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 276
transitive degree	μ_t(G) = 48600

A minimal faithful permutation representation — 2 generators on 276 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);(1,5,3,7)(2,6,4,8)(10,251)(11,250)(12,249)(13,248)(14,247)(15,246)(16,245)(17,244)(18,243)(19,242)(20,241)(21,240)(22,239)(23,238)(24,237)(25,236)(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)(253,276)(254,275)(255,274)(256,273)(257,272)(258,271)(259,270)(260,269)(261,268)(262,267)(263,266)(264,265)

Polycyclic presentation:

2^3,3,5,3,5,3^3|a^2=c,b^2=c,d^3=f,e^5=g,f^3=h,h^3=i,i^3=j,b^a=bc,d^a=d^2f^2h^2i^2j^2,e^a=e^4g^4,f^a=f^2h^2i^2j^2,g^a=g^4,h^a=h^2i^2j^2,i^a=i^2j^2,j^a=j^2

Conjugacy classes & character table

12153 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", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 3, 5, 3, 3, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2*F.8^2*F.9^2*F.10^2);;
gap> SetConjugate(coll, 5, 1, F.5^4*F.7^4);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.8^2*F.9^2*F.10^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2*F.10^2);;
gap> SetConjugate(coll, 9, 1, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12150 #72282005