Dic₅₀₃₂ #72216623

order	20128
automorphism	Aut(G) order 46374912
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₀₃₂

Splittings

Hall	C₁₇×C₃₇ ⋊ Q₃₂
Sylow	C₃₇ ⋊ Dic₁₃₆
Sylow	C₁₇ ⋊ Dic₂₉₆

Subgroups

	subgroup	G/H
center	C₂	D₅₀₃₂
commutator, cosocle	C₈×C₁₇×C₃₇	C₂²
Frattini	C₈	C₂×D₆₂₉
Fitting	C₁₆×C₁₇×C₃₇	C₂
socle, prime radical	C₂×C₁₇×C₃₇	D₈
hypercenter	C₁₆	D₆₂₉
hypocenter	C₁₇×C₃₇	Q₃₂
LCS	G ▷ C₈×C₁₇×C₃₇ ▷ C₄×C₁₇×C₃₇ ▷ C₂×C₁₇×C₃₇ ▷ C₁₇×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 86
transitive degree	μ_t(G) = 20128

A minimal faithful permutation representation — 2 generators on 86 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);(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)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Polycyclic presentation:

2^5,17,37|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^16,g^a=g^36

Conjugacy classes & character table

5035 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₀₆₄, Dic₁₅₀₉₆
maximal normal subgroup of	Dic₁₀₀₆₄
maximal quotient of	Dic₁₅₀₉₆

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, 2, 2, 17, 37]);;
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.6^16);;
gap> SetConjugate(coll, 7, 1, F.7^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5032 #72216623