Dic₁₅₇₃₂ #72295573

order	62928
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₇₃₂

Splittings

Hall	C₉×C₁₉×C₂₃ ⋊ Q₁₆
Hall	C₁₉×C₂₃ ⋊ Dic₃₆
Hall	C₉×C₂₃ ⋊ Dic₇₆
Hall	C₉×C₁₉ ⋊ Dic₉₂
Sylow	C₂₃ ⋊ Dic₆₈₄
Sylow	C₁₉ ⋊ Dic₈₂₈
Sylow	C₉ ⋊ Dic₁₇₄₈

Subgroups

	subgroup	G/H
center	C₂	D₁₅₇₃₂
commutator, cosocle	C₄×C₉×C₁₉×C₂₃	C₂²
Frattini	C₃×C₄	C₂×D₁₃₁₁
Fitting	C₈×C₉×C₁₉×C₂₃	C₂
socle, prime radical	C₂×C₃×C₁₉×C₂₃	D₁₂
hypercenter	C₈	D₃₉₃₃
hypocenter	C₉×C₁₉×C₂₃	Q₁₆
LCS	G ▷ C₄×C₉×C₁₉×C₂₃ ▷ C₂×C₉×C₁₉×C₂₃ ▷ C₉×C₁₉×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 62928

A minimal faithful permutation representation — 2 generators on 67 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)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,25)(19,24)(20,23)(21,22)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2^4,3,19,23,3|a^2=d,b^2=cd,c^2=d,e^3=h,b^a=bc,c^a=cd,e^a=e^2h^2,f^a=f^18,g^a=g^22,h^a=h^2

Conjugacy classes & character table

15735 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, 3, 19, 23, 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.8);;
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.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^22);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15732 #72295573