Dic₁₄₁₅₇ #72290638

order	56628
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₁₅₇

Splittings

derived, Hall	C₉×C₁₃×C₁₂₁ ⋊ C₄
Hall	C₁₃×C₁₂₁ ⋊ Dic₉
Hall	C₉×C₁₂₁ ⋊ Dic₁₃
Sylow	C₁₂₁ ⋊ Dic₁₁₇
Hall	C₉×C₁₃ ⋊ Dic₁₂₁
Sylow	C₁₃ ⋊ Dic₁₀₈₉
Sylow	C₉ ⋊ Dic₁₅₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₄₁₅₇
commutator, hypocenter	C₉×C₁₃×C₁₂₁	C₄
Frattini	C₂×C₃×C₁₁	D₄₂₉
Fitting, cosocle	C₂×C₉×C₁₃×C₁₂₁	C₂
socle, prime radical	C₂×C₃×C₁₁×C₁₃	D₃₃
LCS	G ▷ C₉×C₁₃×C₁₂₁
UCS	C₁ ◁ C₂

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 147
transitive degree	μ_t(G) = 56628

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

Polycyclic presentation:

2^2,3,11,13,3,11|a^2=b,c^3=f,d^11=g,c^a=c^2f^2,d^a=d^10g^10,e^a=e^12,f^a=f^2,g^a=g^10

Conjugacy classes & character table

14160 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");;
gap> coll := SingleCollector(F, [2, 2, 3, 11, 13, 3, 11]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.6^2);;
gap> SetConjugate(coll, 4, 1, F.4^10*F.7^10);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14157 #72290638