Dic₄₃₉₂ #72215841

order	17568
automorphism	Aut(G) order 25297920
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₃₉₂

Splittings

Hall	C₉×C₆₁ ⋊ Q₃₂
Sylow	C₆₁ ⋊ Dic₇₂
Sylow	C₉ ⋊ Dic₄₈₈

Subgroups

	subgroup	G/H
center	C₂	D₄₃₉₂
commutator, cosocle	C₈×C₉×C₆₁	C₂²
Frattini	C₃×C₈	C₂×D₁₈₃
Fitting	C₉×C₁₆×C₆₁	C₂
socle, prime radical	C₂×C₃×C₆₁	D₂₄
hypercenter	C₁₆	D₅₄₉
hypocenter	C₉×C₆₁	Q₃₂
LCS	G ▷ C₈×C₉×C₆₁ ▷ C₄×C₉×C₆₁ ▷ C₂×C₉×C₆₁ ▷ C₉×C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 102
transitive degree	μ_t(G) = 17568

A minimal faithful permutation representation — 2 generators on 102 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)(42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102);(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)(34,41)(35,40)(36,39)(37,38)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)

Polycyclic presentation:

2^5,3,61,3|a^2=e,b^2=cd,c^2=de,d^2=e,f^3=h,b^a=bc,c^a=cd,d^a=de,f^a=f^2h^2,g^a=g^60,h^a=h^2

Conjugacy classes & character table

4395 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₇₈₄, Dic₁₃₁₇₆
maximal normal subgroup of	Dic₈₇₈₄
maximal quotient of	Dic₁₃₁₇₆

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, 3, 61, 3]);;
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.6^2*F.8^2);;
gap> SetConjugate(coll, 7, 1, F.7^60);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4392 #72215841