Dic₁₁₇₀₄ #72277797

order	46816
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₁₇₀₄

Splittings

Hall	C₇×C₁₁×C₁₉ ⋊ Q₃₂
Hall	C₁₁×C₁₉ ⋊ Dic₅₆
Hall	C₇×C₁₉ ⋊ Dic₈₈
Hall	C₇×C₁₁ ⋊ Dic₁₅₂
Sylow	C₁₉ ⋊ Dic₆₁₆
Sylow	C₁₁ ⋊ Dic₁₀₆₄
Sylow	C₇ ⋊ Dic₁₆₇₂

Subgroups

	subgroup	G/H
center	C₂	D₁₁₇₀₄
commutator, cosocle	C₇×C₈×C₁₁×C₁₉	C₂²
Frattini	C₈	C₂×D₁₄₆₃
Fitting	C₇×C₁₁×C₁₆×C₁₉	C₂
socle, prime radical	C₂×C₇×C₁₁×C₁₉	D₈
hypercenter	C₁₆	D₁₄₆₃
hypocenter	C₇×C₁₁×C₁₉	Q₃₂
LCS	G ▷ C₇×C₈×C₁₁×C₁₉ ▷ C₄×C₇×C₁₁×C₁₉ ▷ C₂×C₇×C₁₁×C₁₉ ▷ C₇×C₁₁×C₁₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 46816

A minimal faithful permutation representation — 2 generators on 69 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);(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)(34,39)(35,38)(36,37)(41,50)(42,49)(43,48)(44,47)(45,46)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)

Polycyclic presentation:

2^5,7,11,19|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^6,g^a=g^10,h^a=h^18

Conjugacy classes & character table

11707 conjugacy classes. The character table is too large to display.

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, 19]);;
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> 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);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> SetConjugate(coll, 8, 1, F.8^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic11704 #72277797