Dic₁₃₅₆₃ #72289673

order	54252
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₅₆₃

Splittings

derived, Hall	C₉×C₁₁×C₁₃₇ ⋊ C₄
Hall	C₁₁×C₁₃₇ ⋊ Dic₉
Hall	C₉×C₁₃₇ ⋊ Dic₁₁
Sylow	C₁₃₇ ⋊ Dic₉₉
Hall	C₉×C₁₁ ⋊ Dic₁₃₇
Sylow	C₁₁ ⋊ Dic₁₂₃₃
Sylow	C₉ ⋊ Dic₁₅₀₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₃₅₆₃
commutator, hypocenter	C₉×C₁₁×C₁₃₇	C₄
Frattini	C₂×C₃	D₄₅₂₁
Fitting, cosocle	C₂×C₉×C₁₁×C₁₃₇	C₂
socle, prime radical	C₂×C₃×C₁₁×C₁₃₇	D₃
LCS	G ▷ C₉×C₁₁×C₁₃₇
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 161
transitive degree	μ_t(G) = 54252

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

Polycyclic presentation:

2^2,3,11,137,3|a^2=b,c^3=f,c^a=c^2f^2,d^a=d^10,e^a=e^136,f^a=f^2

Conjugacy classes & character table

13566 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, 11, 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.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^136);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13563 #72289673