Dic₁₆₂₇₂ #72301446

order	65088
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₆₂₇₂

Splittings

Hall	C₉×C₁₁₃ ⋊ Q₆₄
Sylow	C₁₁₃ ⋊ Dic₁₄₄
Sylow	C₉ ⋊ Dic₁₈₀₈

Subgroups

	subgroup	G/H
center	C₂	D₁₆₂₇₂
commutator, cosocle	C₉×C₁₆×C₁₁₃	C₂²
Frattini	C₃×C₁₆	C₂×D₃₃₉
Fitting	C₉×C₃₂×C₁₁₃	C₂
socle, prime radical	C₂×C₃×C₁₁₃	D₄₈
hypercenter	C₃₂	D₁₀₁₇
hypocenter	C₉×C₁₁₃	Q₆₄
LCS	G ▷ C₉×C₁₆×C₁₁₃ ▷ C₈×C₉×C₁₁₃ ▷ C₄×C₉×C₁₁₃ ▷ C₂×C₉×C₁₁₃ ▷ C₉×C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 186
transitive degree	μ_t(G) = 65088

A minimal faithful permutation representation — 2 generators on 186 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)(33,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34)(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);(1,33,17,49)(2,34,18,50)(3,35,19,51)(4,36,20,52)(5,37,21,53)(6,38,22,54)(7,39,23,55)(8,40,24,56)(9,41,25,57)(10,42,26,58)(11,43,27,59)(12,44,28,60)(13,45,29,61)(14,46,30,62)(15,47,31,63)(16,48,32,64)(66,73)(67,72)(68,71)(69,70)(75,186)(76,185)(77,184)(78,183)(79,182)(80,181)(81,180)(82,179)(83,178)(84,177)(85,176)(86,175)(87,174)(88,173)(89,172)(90,171)(91,170)(92,169)(93,168)(94,167)(95,166)(96,165)(97,164)(98,163)(99,162)(100,161)(101,160)(102,159)(103,158)(104,157)(105,156)(106,155)(107,154)(108,153)(109,152)(110,151)(111,150)(112,149)(113,148)(114,147)(115,146)(116,145)(117,144)(118,143)(119,142)(120,141)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)

Polycyclic presentation:

2^6,3,113,3|a^2=f,b^2=cd,c^2=de,d^2=ef,e^2=f,g^3=i,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2i^2,h^a=h^112,i^a=i^2

Conjugacy classes & character table

16275 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", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 113, 3]);;
gap> SetPower(coll, 1, F.6);;
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);;
gap> SetPower(coll, 7, F.9);;
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, 7, 1, F.7^2*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^112);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic16272 #72301446