Dic₄₁₀₄ #72215464

order	16416
automorphism	Aut(G) order 21275136
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₁₀₄

Splittings

Hall	C₁₉×C₂₇ ⋊ Q₃₂
Sylow	C₂₇ ⋊ Dic₁₅₂
Sylow	C₁₉ ⋊ Dic₂₁₆

Subgroups

	subgroup	G/H
center	C₂	D₄₁₀₄
commutator, cosocle	C₈×C₁₉×C₂₇	C₂²
Frattini	C₈×C₉	C₂×D₅₇
Fitting	C₁₆×C₁₉×C₂₇	C₂
socle, prime radical	C₂×C₃×C₁₉	D₇₂
hypercenter	C₁₆	D₅₁₃
hypocenter	C₁₉×C₂₇	Q₃₂
LCS	G ▷ C₈×C₁₉×C₂₇ ▷ C₄×C₁₉×C₂₇ ▷ C₂×C₁₉×C₂₇ ▷ C₁₉×C₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 78
transitive degree	μ_t(G) = 16416

A minimal faithful permutation representation — 2 generators on 78 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);(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)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)

Polycyclic presentation:

2^5,3,19,3^2|a^2=e,b^2=cd,c^2=de,d^2=e,f^3=h,h^3=i,b^a=bc,c^a=cd,d^a=de,f^a=f^2h^2i^2,g^a=g^18,h^a=h^2i^2,i^a=i^2

Conjugacy classes & character table

4107 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₂₀₈, Dic₁₂₃₁₂
maximal normal subgroup of	Dic₈₂₀₈
maximal quotient of	Dic₁₂₃₁₂

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 19, 3, 3]);;
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> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
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*F.8^2*F.9^2);;
gap> SetConjugate(coll, 7, 1, F.7^18);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4104 #72215464