Dic₁₆₁₀₄ #72299992

order	64416
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₆₁₀₄

Splittings

Hall	C₃×C₁₁×C₆₁ ⋊ Q₃₂
Hall	C₁₁×C₆₁ ⋊ Dic₂₄
Hall	C₃×C₆₁ ⋊ Dic₈₈
Sylow	C₆₁ ⋊ Dic₂₆₄
Hall	C₃×C₁₁ ⋊ Dic₄₈₈
Sylow	C₁₁ ⋊ Dic₁₄₆₄
Sylow	C₃ ⋊ Dic₅₃₆₈

Subgroups

	subgroup	G/H
center	C₂	D₁₆₁₀₄
commutator, cosocle	C₃×C₈×C₁₁×C₆₁	C₂²
Frattini	C₈	C₂×D₂₀₁₃
Fitting	C₃×C₁₁×C₁₆×C₆₁	C₂
socle, prime radical	C₂×C₃×C₁₁×C₆₁	D₈
hypercenter	C₁₆	D₂₀₁₃
hypocenter	C₃×C₁₁×C₆₁	Q₃₂
LCS	G ▷ C₃×C₈×C₁₁×C₆₁ ▷ C₃×C₄×C₁₁×C₆₁ ▷ C₂×C₃×C₁₁×C₆₁ ▷ C₃×C₁₁×C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 64416

A minimal faithful permutation representation — 2 generators on 107 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,35)(37,46)(38,45)(39,44)(40,43)(41,42)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)

Polycyclic presentation:

2^5,3,11,61|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^10,h^a=h^60

Conjugacy classes & character table

16107 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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 11, 61]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> SetConjugate(coll, 8, 1, F.8^60);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic16104 #72299992