Dic₅₂₅₂ #72217023

order	21008
automorphism	Aut(G) order 50419200
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₂₅₂

Splittings

Hall	C₁₃×C₁₀₁ ⋊ Q₁₆
Sylow	C₁₀₁ ⋊ Dic₅₂
Sylow	C₁₃ ⋊ Dic₄₀₄

Subgroups

	subgroup	G/H
center	C₂	D₅₂₅₂
commutator, cosocle	C₄×C₁₃×C₁₀₁	C₂²
Frattini	C₄	C₂×D₁₃₁₃
Fitting	C₈×C₁₃×C₁₀₁	C₂
socle, prime radical	C₂×C₁₃×C₁₀₁	D₄
hypercenter	C₈	D₁₃₁₃
hypocenter	C₁₃×C₁₀₁	Q₁₆
LCS	G ▷ C₄×C₁₃×C₁₀₁ ▷ C₂×C₁₃×C₁₀₁ ▷ C₁₃×C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 130
transitive degree	μ_t(G) = 21008

A minimal faithful permutation representation — 2 generators on 130 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)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(31,130)(32,129)(33,128)(34,127)(35,126)(36,125)(37,124)(38,123)(39,122)(40,121)(41,120)(42,119)(43,118)(44,117)(45,116)(46,115)(47,114)(48,113)(49,112)(50,111)(51,110)(52,109)(53,108)(54,107)(55,106)(56,105)(57,104)(58,103)(59,102)(60,101)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)

Polycyclic presentation:

2^4,13,101|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^12,f^a=f^100

Conjugacy classes & character table

5255 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₅₀₄, Dic₁₅₇₅₆
maximal normal subgroup of	Dic₁₀₅₀₄
maximal quotient of	Dic₁₅₇₅₆

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 13, 101]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^100);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5252 #72217023