Dic₆₅₄ #72210596

order	2616
automorphism	Aut(G) order 565056
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₅₄

Splittings

Hall	C₃×C₁₀₉ ⋊ Q₈
Sylow	C₁₀₉ ⋊ Dic₆
Sylow	C₃ ⋊ Dic₂₁₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₂₇
commutator, socle, prime radical, cosocle	C₂×C₃×C₁₀₉	C₂²
Fitting	C₃×C₄×C₁₀₉	C₂
hypercenter	C₄	D₃₂₇
hypocenter	C₃×C₁₀₉	Q₈
LCS	G ▷ C₂×C₃×C₁₀₉ ▷ C₃×C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 120
transitive degree	μ_t(G) = 2616

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

Polycyclic presentation:

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

Conjugacy classes & character table

657 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₁₃₀₈, Dic₁₉₆₂, Dic₃₂₇₀, …
maximal normal subgroup of	Dic₁₃₀₈
maximal quotient of	Dic₁₉₆₂, Dic₃₂₇₀, Dic₄₅₇₈, …

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, 3, 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.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^108);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic654 #72210596