Dic₁₃₀₆₈ #72288640

order	52272
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₀₆₈

Splittings

Hall	C₂₇×C₁₂₁ ⋊ Q₁₆
Sylow	C₁₂₁ ⋊ Dic₁₀₈
Sylow	C₂₇ ⋊ Dic₄₈₄

Subgroups

	subgroup	G/H
center	C₂	D₁₃₀₆₈
commutator, cosocle	C₄×C₂₇×C₁₂₁	C₂²
Frattini	C₄×C₉×C₁₁	C₂×D₃₃
Fitting	C₈×C₂₇×C₁₂₁	C₂
socle, prime radical	C₂×C₃×C₁₁	D₃₉₆
hypercenter	C₈	D₃₂₆₇
hypocenter	C₂₇×C₁₂₁	Q₁₆
LCS	G ▷ C₄×C₂₇×C₁₂₁ ▷ C₂×C₂₇×C₁₂₁ ▷ C₂₇×C₁₂₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 164
transitive degree	μ_t(G) = 52272

A minimal faithful permutation representation — 2 generators on 164 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(45,164)(46,163)(47,162)(48,161)(49,160)(50,159)(51,158)(52,157)(53,156)(54,155)(55,154)(56,153)(57,152)(58,151)(59,150)(60,149)(61,148)(62,147)(63,146)(64,145)(65,144)(66,143)(67,142)(68,141)(69,140)(70,139)(71,138)(72,137)(73,136)(74,135)(75,134)(76,133)(77,132)(78,131)(79,130)(80,129)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,120)(90,119)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105)

Polycyclic presentation:

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

Conjugacy classes & character table

13071 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, 3, 11, 3, 11, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
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, 5, 1, F.5^2*F.7^2*F.9^2);;
gap> SetConjugate(coll, 6, 1, F.6^10*F.8^10);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^10);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13068 #72288640