D₁₁₀₉₇ #72215838

order	22194
automorphism	Aut(G) order 81496368
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₁₁₀₉₇

Splittings

Fitting, derived, Hall	C₈₁×C₁₃₇ ⋊ C₂
Sylow	C₁₃₇ ⋊ D₈₁
Sylow	C₈₁ ⋊ D₁₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₈₁×C₁₃₇	C₂
Frattini	C₂₇	D₄₁₁
socle	C₃×C₁₃₇	D₂₇
LCS	G ▷ C₈₁×C₁₃₇
UCS	C₁

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 218
transitive degree	μ_t(G) = 11097

A minimal faithful permutation representation — 2 generators on 218 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);(2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(83,218)(84,217)(85,216)(86,215)(87,214)(88,213)(89,212)(90,211)(91,210)(92,209)(93,208)(94,207)(95,206)(96,205)(97,204)(98,203)(99,202)(100,201)(101,200)(102,199)(103,198)(104,197)(105,196)(106,195)(107,194)(108,193)(109,192)(110,191)(111,190)(112,189)(113,188)(114,187)(115,186)(116,185)(117,184)(118,183)(119,182)(120,181)(121,180)(122,179)(123,178)(124,177)(125,176)(126,175)(127,174)(128,173)(129,172)(130,171)(131,170)(132,169)(133,168)(134,167)(135,166)(136,165)(137,164)(138,163)(139,162)(140,161)(141,160)(142,159)(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)

Polycyclic presentation:

2,3,137,3^3|b^3=d,d^3=e,e^3=f,b^a=b^2d^2e^2f^2,c^a=c^136,d^a=d^2e^2f^2,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

5550 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₁₀₉₇
maximal normal subgroup of	C₂×D₁₁₀₉₇
maximal quotient of	Dic₁₁₀₉₇, C₂×D₁₁₀₉₇

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");;
gap> coll := SingleCollector(F, [2, 3, 137, 3, 3, 3]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.4^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^136);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11097 #72215838