Dic₁₆₀₂₉ #72298451

order	64116
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₆₀₂₉

Splittings

derived, Hall	C₉×C₁₃×C₁₃₇ ⋊ C₄
Hall	C₁₃×C₁₃₇ ⋊ Dic₉
Hall	C₉×C₁₃₇ ⋊ Dic₁₃
Sylow	C₁₃₇ ⋊ Dic₁₁₇
Hall	C₉×C₁₃ ⋊ Dic₁₃₇
Sylow	C₁₃ ⋊ Dic₁₂₃₃
Sylow	C₉ ⋊ Dic₁₇₈₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₆₀₂₉
commutator, hypocenter	C₉×C₁₃×C₁₃₇	C₄
Frattini	C₂×C₃	D₅₃₄₃
Fitting, cosocle	C₂×C₉×C₁₃×C₁₃₇	C₂
socle, prime radical	C₂×C₃×C₁₃×C₁₃₇	D₃
LCS	G ▷ C₉×C₁₃×C₁₃₇
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 163
transitive degree	μ_t(G) = 64116

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

Polycyclic presentation:

2^2,3,13,137,3|a^2=b,c^3=f,c^a=c^2f^2,d^a=d^12,e^a=e^136,f^a=f^2

Conjugacy classes & character table

16032 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");;
gap> coll := SingleCollector(F, [2, 2, 3, 13, 137, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.6^2);;
gap> SetConjugate(coll, 4, 1, F.4^12);;
gap> SetConjugate(coll, 5, 1, F.5^136);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic16029 #72298451