Dic₃₂₆₀ #72213640

order	13040
automorphism	Aut(G) order 16899840
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₂₆₀

Splittings

Hall	C₅×C₁₆₃ ⋊ Q₁₆
Sylow	C₁₆₃ ⋊ Dic₂₀
Sylow	C₅ ⋊ Dic₆₅₂

Subgroups

	subgroup	G/H
center	C₂	D₃₂₆₀
commutator, cosocle	C₄×C₅×C₁₆₃	C₂²
Frattini	C₄	C₂×D₈₁₅
Fitting	C₅×C₈×C₁₆₃	C₂
socle, prime radical	C₂×C₅×C₁₆₃	D₄
hypercenter	C₈	D₈₁₅
hypocenter	C₅×C₁₆₃	Q₁₆
LCS	G ▷ C₄×C₅×C₁₆₃ ▷ C₂×C₅×C₁₆₃ ▷ C₅×C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 184
transitive degree	μ_t(G) = 13040

A minimal faithful permutation representation — 2 generators on 184 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,21)(19,20)(23,184)(24,183)(25,182)(26,181)(27,180)(28,179)(29,178)(30,177)(31,176)(32,175)(33,174)(34,173)(35,172)(36,171)(37,170)(38,169)(39,168)(40,167)(41,166)(42,165)(43,164)(44,163)(45,162)(46,161)(47,160)(48,159)(49,158)(50,157)(51,156)(52,155)(53,154)(54,153)(55,152)(56,151)(57,150)(58,149)(59,148)(60,147)(61,146)(62,145)(63,144)(64,143)(65,142)(66,141)(67,140)(68,139)(69,138)(70,137)(71,136)(72,135)(73,134)(74,133)(75,132)(76,131)(77,130)(78,129)(79,128)(80,127)(81,126)(82,125)(83,124)(84,123)(85,122)(86,121)(87,120)(88,119)(89,118)(90,117)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)

Polycyclic presentation:

2^4,5,163|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^162

Conjugacy classes & character table

3263 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₅₂₀, Dic₉₇₈₀, Dic₁₆₃₀₀
maximal normal subgroup of	Dic₆₅₂₀
maximal quotient of	Dic₉₇₈₀, Dic₁₆₃₀₀

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 5, 163]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^162);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3260 #72213640