Dic₃₆₃₆ #72214710

order	14544
automorphism	Aut(G) order 17452800
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₆₃₆

Splittings

Hall	C₉×C₁₀₁ ⋊ Q₁₆
Sylow	C₁₀₁ ⋊ Dic₃₆
Sylow	C₉ ⋊ Dic₄₀₄

Subgroups

	subgroup	G/H
center	C₂	D₃₆₃₆
commutator, cosocle	C₄×C₉×C₁₀₁	C₂²
Frattini	C₃×C₄	C₂×D₃₀₃
Fitting	C₈×C₉×C₁₀₁	C₂
socle, prime radical	C₂×C₃×C₁₀₁	D₁₂
hypercenter	C₈	D₉₀₉
hypocenter	C₉×C₁₀₁	Q₁₆
LCS	G ▷ C₄×C₉×C₁₀₁ ▷ C₂×C₉×C₁₀₁ ▷ C₉×C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 126
transitive degree	μ_t(G) = 14544

A minimal faithful permutation representation — 2 generators on 126 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,25)(19,24)(20,23)(21,22)(27,126)(28,125)(29,124)(30,123)(31,122)(32,121)(33,120)(34,119)(35,118)(36,117)(37,116)(38,115)(39,114)(40,113)(41,112)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)

Polycyclic presentation:

2^4,3,101,3|a^2=d,b^2=cd,c^2=d,e^3=g,b^a=bc,c^a=cd,e^a=e^2g^2,f^a=f^100,g^a=g^2

Conjugacy classes & character table

3639 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₂₇₂, Dic₁₀₉₀₈
maximal normal subgroup of	Dic₇₂₇₂
maximal quotient of	Dic₁₀₉₀₈

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 101, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.6^100);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3636 #72214710