Dic₁₅₂₆ #72212003

order	6104
automorphism	Aut(G) order 3955392
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₂₆

Splittings

Hall	C₇×C₁₀₉ ⋊ Q₈
Sylow	C₁₀₉ ⋊ Dic₁₄
Sylow	C₇ ⋊ Dic₂₁₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₆₃
commutator, socle, prime radical, cosocle	C₂×C₇×C₁₀₉	C₂²
Fitting	C₄×C₇×C₁₀₉	C₂
hypercenter	C₄	D₇₆₃
hypocenter	C₇×C₁₀₉	Q₈
LCS	G ▷ C₂×C₇×C₁₀₉ ▷ C₇×C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 124
transitive degree	μ_t(G) = 6104

A minimal faithful permutation representation — 2 generators on 124 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124);(1,5,3,7)(2,6,4,8)(10,15)(11,14)(12,13)(17,124)(18,123)(19,122)(20,121)(21,120)(22,119)(23,118)(24,117)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)

Polycyclic presentation:

2^3,7,109|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^108

Conjugacy classes & character table

1529 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₃₀₅₂, Dic₄₅₇₈, Dic₇₆₃₀, …
maximal normal subgroup of	Dic₃₀₅₂
maximal quotient of	Dic₄₅₇₈, Dic₇₆₃₀, Dic₁₀₆₈₂

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 7, 109]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^108);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1526 #72212003