Dic₁₅₇₆₀ #72295622

order	63040
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₇₆₀

Splittings

Hall	C₅×C₁₉₇ ⋊ Q₆₄
Sylow	C₁₉₇ ⋊ Dic₈₀
Sylow	C₅ ⋊ Dic₃₁₅₂

Subgroups

	subgroup	G/H
center	C₂	D₁₅₇₆₀
commutator, cosocle	C₅×C₁₆×C₁₉₇	C₂²
Frattini	C₁₆	C₂×D₉₈₅
Fitting	C₅×C₃₂×C₁₉₇	C₂
socle, prime radical	C₂×C₅×C₁₉₇	D₁₆
hypercenter	C₃₂	D₉₈₅
hypocenter	C₅×C₁₉₇	Q₆₄
LCS	G ▷ C₅×C₁₆×C₁₉₇ ▷ C₅×C₈×C₁₉₇ ▷ C₄×C₅×C₁₉₇ ▷ C₂×C₅×C₁₉₇ ▷ C₅×C₁₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 266
transitive degree	μ_t(G) = 63040

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

Polycyclic presentation:

2^6,5,197|a^2=f,b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^4,h^a=h^196

Conjugacy classes & character table

15763 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 5, 197]);;
gap> SetPower(coll, 1, F.6);;
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);;
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, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^196);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15760 #72295622