Dic₇₈₄ #72210871

order	3136
automorphism	Aut(G) order 1053696
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₈₄

Splittings

Sylow

C₄₉ ⋊ Q₆₄

Subgroups

	subgroup	G/H
center	C₂	D₇₈₄
commutator, cosocle	C₁₆×C₄₉	C₂²
Frattini	C₇×C₁₆	C₂×D₇
Fitting	C₃₂×C₄₉	C₂
socle, prime radical	C₂×C₇	D₁₁₂
hypercenter	C₃₂	D₄₉
hypocenter	C₄₉	Q₆₄
LCS	G ▷ C₁₆×C₄₉ ▷ C₈×C₄₉ ▷ C₄×C₄₉ ▷ C₂×C₄₉ ▷ C₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

45 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 3136

A minimal faithful permutation representation — 2 generators on 113 points:

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

Polycyclic presentation:

2^6,7^2|a^2=f,b^2=cd,c^2=de,d^2=ef,e^2=f,g^7=h,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^6h^6,h^a=h^6

Conjugacy classes & character table

787 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₁₅₆₈, Dic₂₃₅₂, Dic₃₉₂₀, …
maximal normal subgroup of	Dic₁₅₆₈
maximal quotient of	Dic₂₃₅₂, Dic₃₉₂₀, Dic₅₄₈₈, …

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 7, 7]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 7, F.8);;
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, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^6*F.8^6);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic784 #72210871