D₂₉₅₆₅ #72293553

order	59130
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₉₅₆₅

Splittings

Fitting, derived, Hall	C₅×C₇₃×C₈₁ ⋊ C₂
Hall	C₇₃×C₈₁ ⋊ D₅
Hall	C₅×C₈₁ ⋊ D₇₃
Hall	C₅×C₇₃ ⋊ D₈₁
Sylow	C₈₁ ⋊ D₃₆₅
Sylow	C₇₃ ⋊ D₄₀₅
Sylow	C₅ ⋊ D₅₉₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₇₃×C₈₁	C₂
Frattini	C₂₇	D₁₀₉₅
socle	C₃×C₅×C₇₃	D₂₇
LCS	G ▷ C₅×C₇₃×C₈₁
UCS	C₁

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 159
transitive degree	μ_t(G) = 29565

A minimal faithful permutation representation — 2 generators on 159 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);(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)(83,86)(84,85)(88,159)(89,158)(90,157)(91,156)(92,155)(93,154)(94,153)(95,152)(96,151)(97,150)(98,149)(99,148)(100,147)(101,146)(102,145)(103,144)(104,143)(105,142)(106,141)(107,140)(108,139)(109,138)(110,137)(111,136)(112,135)(113,134)(114,133)(115,132)(116,131)(117,130)(118,129)(119,128)(120,127)(121,126)(122,125)(123,124)

Polycyclic presentation:

2,3,5,73,3^3|b^3=e,e^3=f,f^3=g,b^a=b^2e^2f^2g^2,c^a=c^4,d^a=d^72,e^a=e^2f^2g^2,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

14784 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");;
gap> coll := SingleCollector(F, [2, 3, 5, 73, 3, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^72);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D29565 #72293553