Dic₁₃₆₁₂ #72289760

order	54448
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₆₁₂

Splittings

Hall	C₄₁×C₈₃ ⋊ Q₁₆
Sylow	C₈₃ ⋊ Dic₁₆₄
Sylow	C₄₁ ⋊ Dic₃₃₂

Subgroups

	subgroup	G/H
center	C₂	D₁₃₆₁₂
commutator, cosocle	C₄×C₄₁×C₈₃	C₂²
Frattini	C₄	C₂×D₃₄₀₃
Fitting	C₈×C₄₁×C₈₃	C₂
socle, prime radical	C₂×C₄₁×C₈₃	D₄
hypercenter	C₈	D₃₄₀₃
hypocenter	C₄₁×C₈₃	Q₁₆
LCS	G ▷ C₄×C₄₁×C₈₃ ▷ C₂×C₄₁×C₈₃ ▷ C₄₁×C₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 140
transitive degree	μ_t(G) = 54448

A minimal faithful permutation representation — 2 generators on 140 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(59,140)(60,139)(61,138)(62,137)(63,136)(64,135)(65,134)(66,133)(67,132)(68,131)(69,130)(70,129)(71,128)(72,127)(73,126)(74,125)(75,124)(76,123)(77,122)(78,121)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)

Polycyclic presentation:

2^4,41,83|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^40,f^a=f^82

Conjugacy classes & character table

13615 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, 2, 2, 41, 83]);;
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.5^40);;
gap> SetConjugate(coll, 6, 1, F.6^82);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13612 #72289760