Dic₆₁₂ #72210474

order	2448
automorphism	Aut(G) order 470016
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₁₂

Splittings

Hall	C₉×C₁₇ ⋊ Q₁₆
Sylow	C₁₇ ⋊ Dic₃₆
Sylow	C₉ ⋊ Dic₆₈

Subgroups

	subgroup	G/H
center	C₂	D₆₁₂
commutator, cosocle	C₄×C₉×C₁₇	C₂²
Frattini	C₃×C₄	C₂×D₅₁
Fitting	C₈×C₉×C₁₇	C₂
socle, prime radical	C₂×C₃×C₁₇	D₁₂
hypercenter	C₈	D₁₅₃
hypocenter	C₉×C₁₇	Q₁₆
LCS	G ▷ C₄×C₉×C₁₇ ▷ C₂×C₉×C₁₇ ▷ C₉×C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 2448

A minimal faithful permutation representation — 2 generators on 42 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,25)(19,24)(20,23)(21,22)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)

Polycyclic presentation:

2^4,3,17,3|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^16,g^a=g^2

Conjugacy classes & character table

615 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₁₂₂₄, Dic₁₈₃₆, Dic₃₀₆₀, …
maximal normal subgroup of	Dic₁₂₂₄
maximal quotient of	Dic₁₈₃₆, Dic₃₀₆₀, Dic₄₂₈₄, …

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, 17, 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.6^16);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic612 #72210474