Dic₄₃₁₂ #72215739

order	17248
automorphism	Aut(G) order 28976640
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₃₁₂

Splittings

Hall	C₁₁×C₄₉ ⋊ Q₃₂
Sylow	C₄₉ ⋊ Dic₈₈
Sylow	C₁₁ ⋊ Dic₃₉₂

Subgroups

	subgroup	G/H
center	C₂	D₄₃₁₂
commutator, cosocle	C₈×C₁₁×C₄₉	C₂²
Frattini	C₇×C₈	C₂×D₇₇
Fitting	C₁₁×C₁₆×C₄₉	C₂
socle, prime radical	C₂×C₇×C₁₁	D₅₆
hypercenter	C₁₆	D₅₃₉
hypocenter	C₁₁×C₄₉	Q₃₂
LCS	G ▷ C₈×C₁₁×C₄₉ ▷ C₄×C₁₁×C₄₉ ▷ C₂×C₁₁×C₄₉ ▷ C₁₁×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 92
transitive degree	μ_t(G) = 17248

A minimal faithful permutation representation — 2 generators on 92 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,81)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(83,92)(84,91)(85,90)(86,89)(87,88)

Polycyclic presentation:

2^5,7,11,7|a^2=e,b^2=cd,c^2=de,d^2=e,f^7=h,b^a=bc,c^a=cd,d^a=de,f^a=f^6h^6,g^a=g^10,h^a=h^6

Conjugacy classes & character table

4315 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₆₂₄, Dic₁₂₉₃₆
maximal normal subgroup of	Dic₈₆₂₄
maximal quotient of	Dic₁₂₉₃₆

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, 7, 11, 7]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
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, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^6*F.8^6);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4312 #72215739