Dic₅₉₁₃ #72224232

order	23652
automorphism	Aut(G) order 45979488
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₉₁₃

Splittings

derived, Hall	C₇₃×C₈₁ ⋊ C₄
Sylow	C₈₁ ⋊ Dic₇₃
Sylow	C₇₃ ⋊ Dic₈₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₉₁₃
commutator, hypocenter	C₇₃×C₈₁	C₄
Frattini	C₂×C₂₇	D₂₁₉
Fitting, cosocle	C₂×C₇₃×C₈₁	C₂
socle, prime radical	C₂×C₃×C₇₃	D₂₇
LCS	G ▷ C₇₃×C₈₁
UCS	C₁ ◁ C₂

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 158
transitive degree	μ_t(G) = 23652

A minimal faithful permutation representation — 2 generators on 158 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);(1,3,2,4)(6,85)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(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)

Polycyclic presentation:

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

Conjugacy classes & character table

5916 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₈₂₆
maximal normal subgroup of	Dic₁₁₈₂₆

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, 2, 3, 73, 3, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.5^2*F.6^2*F.7^2);;
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);

Dic5913 #72224232