Dic₅₀₄₄ #72216630

order	20176
automorphism	Aut(G) order 46485504
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₀₄₄

Splittings

Hall	C₁₃×C₉₇ ⋊ Q₁₆
Sylow	C₉₇ ⋊ Dic₅₂
Sylow	C₁₃ ⋊ Dic₃₈₈

Subgroups

	subgroup	G/H
center	C₂	D₅₀₄₄
commutator, cosocle	C₄×C₁₃×C₉₇	C₂²
Frattini	C₄	C₂×D₁₂₆₁
Fitting	C₈×C₁₃×C₉₇	C₂
socle, prime radical	C₂×C₁₃×C₉₇	D₄
hypercenter	C₈	D₁₂₆₁
hypocenter	C₁₃×C₉₇	Q₁₆
LCS	G ▷ C₄×C₁₃×C₉₇ ▷ C₂×C₁₃×C₉₇ ▷ C₁₃×C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 126
transitive degree	μ_t(G) = 20176

A minimal faithful permutation representation — 2 generators on 126 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);(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)(31,126)(32,125)(33,124)(34,123)(35,122)(36,121)(37,120)(38,119)(39,118)(40,117)(41,116)(42,115)(43,114)(44,113)(45,112)(46,111)(47,110)(48,109)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)

Polycyclic presentation:

2^4,13,97|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^12,f^a=f^96

Conjugacy classes & character table

5047 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₀₈₈, Dic₁₅₁₃₂
maximal normal subgroup of	Dic₁₀₀₈₈
maximal quotient of	Dic₁₅₁₃₂

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, 97]);;
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.6^96);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5044 #72216630