Dic₁₇₆₄ #72212191

order	7056
automorphism	Aut(G) order 3556224
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₇₆₄

Splittings

Hall	C₉×C₄₉ ⋊ Q₁₆
Sylow	C₄₉ ⋊ Dic₃₆
Sylow	C₉ ⋊ Dic₁₉₆

Subgroups

	subgroup	G/H
center	C₂	D₁₇₆₄
commutator, cosocle	C₄×C₉×C₄₉	C₂²
Frattini	C₃×C₄×C₇	C₂×D₂₁
Fitting	C₈×C₉×C₄₉	C₂
socle, prime radical	C₂×C₃×C₇	D₈₄
hypercenter	C₈	D₄₄₁
hypocenter	C₉×C₄₉	Q₁₆
LCS	G ▷ C₄×C₉×C₄₉ ▷ C₂×C₉×C₄₉ ▷ C₉×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

81 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 74
transitive degree	μ_t(G) = 7056

A minimal faithful permutation representation — 2 generators on 74 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,25)(19,24)(20,23)(21,22)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Polycyclic presentation:

2^4,3,7,3,7|a^2=d,b^2=cd,c^2=d,e^3=g,f^7=h,b^a=bc,c^a=cd,e^a=e^2g^2,f^a=f^6h^6,g^a=g^2,h^a=h^6

Conjugacy classes & character table

1767 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₃₅₂₈, Dic₅₂₉₂, Dic₈₈₂₀, …
maximal normal subgroup of	Dic₃₅₂₈
maximal quotient of	Dic₅₂₉₂, Dic₈₈₂₀, Dic₁₂₃₄₈

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

Dic1764 #72212191