Dic₁₃₇₀₃ #72289906

order	54812
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₇₀₃

Splittings

derived, Hall	C₇₁×C₁₉₃ ⋊ C₄
Sylow	C₁₉₃ ⋊ Dic₇₁
Sylow	C₇₁ ⋊ Dic₁₉₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₇₀₃
commutator, hypocenter	C₇₁×C₁₉₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇₁×C₁₉₃	C₂
LCS	G ▷ C₇₁×C₁₉₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 268
transitive degree	μ_t(G) = 54812

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

Polycyclic presentation:

2^2,71,193|a^2=b,c^a=c^70,d^a=d^192

Conjugacy classes & character table

13706 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");;
gap> coll := SingleCollector(F, [2, 2, 71, 193]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^70);;
gap> SetConjugate(coll, 4, 1, F.4^192);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13703 #72289906