Dic₄₃₁₆ #72215735

order	17264
automorphism	Aut(G) order 33975552
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₃₁₆

Splittings

Hall	C₁₃×C₈₃ ⋊ Q₁₆
Sylow	C₈₃ ⋊ Dic₅₂
Sylow	C₁₃ ⋊ Dic₃₃₂

Subgroups

	subgroup	G/H
center	C₂	D₄₃₁₆
commutator, cosocle	C₄×C₁₃×C₈₃	C₂²
Frattini	C₄	C₂×D₁₀₇₉
Fitting	C₈×C₁₃×C₈₃	C₂
socle, prime radical	C₂×C₁₃×C₈₃	D₄
hypercenter	C₈	D₁₀₇₉
hypocenter	C₁₃×C₈₃	Q₁₆
LCS	G ▷ C₄×C₁₃×C₈₃ ▷ C₂×C₁₃×C₈₃ ▷ C₁₃×C₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 112
transitive degree	μ_t(G) = 17264

A minimal faithful permutation representation — 2 generators on 112 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2^4,13,83|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^12,f^a=f^82

Conjugacy classes & character table

4319 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₆₃₂, Dic₁₂₉₄₈
maximal normal subgroup of	Dic₈₆₃₂
maximal quotient of	Dic₁₂₉₄₈

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, 13, 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.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^82);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4316 #72215735