Dic₁₅₁₈ #72211995

order	6072
automorphism	Aut(G) order 2671680
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₁₈

Splittings

Hall	C₃×C₁₁×C₂₃ ⋊ Q₈
Hall	C₁₁×C₂₃ ⋊ Dic₆
Hall	C₃×C₂₃ ⋊ Dic₂₂
Hall	C₃×C₁₁ ⋊ Dic₄₆
Sylow	C₂₃ ⋊ Dic₆₆
Sylow	C₁₁ ⋊ Dic₁₃₈
Sylow	C₃ ⋊ Dic₅₀₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₅₉
commutator, socle, prime radical, cosocle	C₂×C₃×C₁₁×C₂₃	C₂²
Fitting	C₃×C₄×C₁₁×C₂₃	C₂
hypercenter	C₄	D₇₅₉
hypocenter	C₃×C₁₁×C₂₃	Q₈
LCS	G ▷ C₂×C₃×C₁₁×C₂₃ ▷ C₃×C₁₁×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 6072

A minimal faithful permutation representation — 2 generators on 45 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);(1,5,3,7)(2,6,4,8)(10,11)(13,22)(14,21)(15,20)(16,19)(17,18)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)

Polycyclic presentation:

2^3,3,11,23|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^10,f^a=f^22

Conjugacy classes & character table

1521 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₃₀₃₆, Dic₄₅₅₄, Dic₇₅₉₀, …
maximal normal subgroup of	Dic₃₀₃₆
maximal quotient of	Dic₄₅₅₄, Dic₇₅₉₀, Dic₁₀₆₂₆

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 11, 23]);;
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.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1518 #72211995