D₁₃₃₆₅ #72235484

order	26730
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₁₃₃₆₅

Splittings

Fitting, derived, Hall	C₅×C₁₁×C₂₄₃ ⋊ C₂
Hall	C₁₁×C₂₄₃ ⋊ D₅
Hall	C₅×C₂₄₃ ⋊ D₁₁
Sylow	C₂₄₃ ⋊ D₅₅
Hall	C₅×C₁₁ ⋊ D₂₄₃
Sylow	C₁₁ ⋊ D₁₂₁₅
Sylow	C₅ ⋊ D₂₆₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₁₁×C₂₄₃	C₂
Frattini	C₈₁	D₁₆₅
socle	C₃×C₅×C₁₁	D₈₁
LCS	G ▷ C₅×C₁₁×C₂₄₃
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 259
transitive degree	μ_t(G) = 13365

A minimal faithful permutation representation — 2 generators on 259 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);(2,243)(3,242)(4,241)(5,240)(6,239)(7,238)(8,237)(9,236)(10,235)(11,234)(12,233)(13,232)(14,231)(15,230)(16,229)(17,228)(18,227)(19,226)(20,225)(21,224)(22,223)(23,222)(24,221)(25,220)(26,219)(27,218)(28,217)(29,216)(30,215)(31,214)(32,213)(33,212)(34,211)(35,210)(36,209)(37,208)(38,207)(39,206)(40,205)(41,204)(42,203)(43,202)(44,201)(45,200)(46,199)(47,198)(48,197)(49,196)(50,195)(51,194)(52,193)(53,192)(54,191)(55,190)(56,189)(57,188)(58,187)(59,186)(60,185)(61,184)(62,183)(63,182)(64,181)(65,180)(66,179)(67,178)(68,177)(69,176)(70,175)(71,174)(72,173)(73,172)(74,171)(75,170)(76,169)(77,168)(78,167)(79,166)(80,165)(81,164)(82,163)(83,162)(84,161)(85,160)(86,159)(87,158)(88,157)(89,156)(90,155)(91,154)(92,153)(93,152)(94,151)(95,150)(96,149)(97,148)(98,147)(99,146)(100,145)(101,144)(102,143)(103,142)(104,141)(105,140)(106,139)(107,138)(108,137)(109,136)(110,135)(111,134)(112,133)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(121,124)(122,123)(245,248)(246,247)(250,259)(251,258)(252,257)(253,256)(254,255)

Polycyclic presentation:

2,3,5,11,3^4|b^3=e,e^3=f,f^3=g,g^3=h,b^a=b^2e^2f^2g^2h^2,c^a=c^4,d^a=d^10,e^a=e^2f^2g^2h^2,f^a=f^2g^2h^2,g^a=g^2h^2,h^a=h^2

Conjugacy classes & character table

6684 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₃₆₅
maximal normal subgroup of	C₂×D₁₃₃₆₅
maximal quotient of	Dic₁₃₃₆₅, C₂×D₁₃₃₆₅

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, 3, 5, 11, 3, 3, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2*F.6^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D13365 #72235484