Dic₁₄₀₂₂ #72290422

order	56088
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₀₂₂

Splittings

Hall	C₉×C₁₉×C₄₁ ⋊ Q₈
Hall	C₁₉×C₄₁ ⋊ Dic₁₈
Hall	C₉×C₄₁ ⋊ Dic₃₈
Hall	C₉×C₁₉ ⋊ Dic₈₂
Sylow	C₄₁ ⋊ Dic₃₄₂
Sylow	C₁₉ ⋊ Dic₇₃₈
Sylow	C₉ ⋊ Dic₁₅₅₈

Subgroups

	subgroup	G/H
center	C₂	C₂×D₇₀₁₁
commutator, cosocle	C₂×C₉×C₁₉×C₄₁	C₂²
Frattini	C₂×C₃	C₂×D₂₃₃₇
Fitting	C₄×C₉×C₁₉×C₄₁	C₂
socle, prime radical	C₂×C₃×C₁₉×C₄₁	C₂×D₃
hypercenter	C₄	D₇₀₁₁
hypocenter	C₉×C₁₉×C₄₁	Q₈
LCS	G ▷ C₂×C₉×C₁₉×C₄₁ ▷ C₉×C₁₉×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 56088

A minimal faithful permutation representation — 2 generators on 77 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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(1,5,3,7)(2,6,4,8)(10,17)(11,16)(12,15)(13,14)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(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)

Polycyclic presentation:

2^3,3,19,41,3|a^2=c,b^2=c,d^3=g,b^a=bc,d^a=d^2g^2,e^a=e^18,f^a=f^40,g^a=g^2

Conjugacy classes & character table

14025 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 19, 41, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^40);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14022 #72290422