Dic₄₁₂₈ #72215480

order	16512
automorphism	Aut(G) order 22192128
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₁₂₈

Splittings

Hall	C₃×C₄₃ ⋊ Q₁₂₈
Sylow	C₄₃ ⋊ Dic₉₆
Sylow	C₃ ⋊ Dic₁₃₇₆

Subgroups

	subgroup	G/H
center	C₂	D₄₁₂₈
commutator, cosocle	C₃×C₃₂×C₄₃	C₂²
Frattini	C₃₂	C₂×D₁₂₉
Fitting	C₃×C₄₃×C₆₄	C₂
socle, prime radical	C₂×C₃×C₄₃	D₃₂
hypercenter	C₆₄	D₁₂₉
hypocenter	C₃×C₄₃	Q₁₂₈
LCS	G ▷ C₃×C₃₂×C₄₃ ▷ C₃×C₁₆×C₄₃ ▷ C₃×C₈×C₄₃ ▷ C₃×C₄×C₄₃ ▷ C₂×C₃×C₄₃ ▷ C₃×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 174
transitive degree	μ_t(G) = 16512

A minimal faithful permutation representation — 2 generators on 174 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)(65,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66)(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);(1,65,33,97)(2,66,34,98)(3,67,35,99)(4,68,36,100)(5,69,37,101)(6,70,38,102)(7,71,39,103)(8,72,40,104)(9,73,41,105)(10,74,42,106)(11,75,43,107)(12,76,44,108)(13,77,45,109)(14,78,46,110)(15,79,47,111)(16,80,48,112)(17,81,49,113)(18,82,50,114)(19,83,51,115)(20,84,52,116)(21,85,53,117)(22,86,54,118)(23,87,55,119)(24,88,56,120)(25,89,57,121)(26,90,58,122)(27,91,59,123)(28,92,60,124)(29,93,61,125)(30,94,62,126)(31,95,63,127)(32,96,64,128)(130,131)(133,174)(134,173)(135,172)(136,171)(137,170)(138,169)(139,168)(140,167)(141,166)(142,165)(143,164)(144,163)(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)

Polycyclic presentation:

2^7,3,43|a^2=g,b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^2,i^a=i^42

Conjugacy classes & character table

4131 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₂₅₆, Dic₁₂₃₈₄
maximal normal subgroup of	Dic₈₂₅₆
maximal quotient of	Dic₁₂₃₈₄

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 43]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^42);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4128 #72215480