Dic₇₁₂₈ #72236474

order	28512
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₁₂₈

Splittings

Hall	C₁₁×C₈₁ ⋊ Q₃₂
Sylow	C₈₁ ⋊ Dic₈₈
Sylow	C₁₁ ⋊ Dic₆₄₈

Subgroups

	subgroup	G/H
center	C₂	D₇₁₂₈
commutator, cosocle	C₈×C₁₁×C₈₁	C₂²
Frattini	C₈×C₂₇	C₂×D₃₃
Fitting	C₁₁×C₁₆×C₈₁	C₂
socle, prime radical	C₂×C₃×C₁₁	D₂₁₆
hypercenter	C₁₆	D₈₉₁
hypocenter	C₁₁×C₈₁	Q₃₂
LCS	G ▷ C₈×C₁₁×C₈₁ ▷ C₄×C₁₁×C₈₁ ▷ C₂×C₁₁×C₈₁ ▷ C₁₁×C₈₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 124
transitive degree	μ_t(G) = 28512

A minimal faithful permutation representation — 2 generators on 124 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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113)(114,115,116,117,118,119,120,121,122,123,124);(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)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(115,124)(116,123)(117,122)(118,121)(119,120)

Polycyclic presentation:

2^5,3,11,3^3|a^2=e,b^2=cd,c^2=de,d^2=e,f^3=h,h^3=i,i^3=j,b^a=bc,c^a=cd,d^a=de,f^a=f^2h^2i^2j^2,g^a=g^10,h^a=h^2i^2j^2,i^a=i^2j^2,j^a=j^2

Conjugacy classes & character table

7131 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₂₅₆
maximal normal subgroup of	Dic₁₄₂₅₆

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", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 11, 3, 3, 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> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
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*F.9^2*F.10^2);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2*F.10^2);;
gap> SetConjugate(coll, 9, 1, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic7128 #72236474