D₂₀₁₇₆ #72274346

order	40352
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₀₁₇₆

Splittings

Fitting	C₁₃×C₁₆×C₉₇ ⋊ C₂
Hall	C₁₃×C₉₇ ⋊ D₁₆
Sylow	C₉₇ ⋊ D₂₀₈
Sylow	C₁₃ ⋊ D₁₅₅₂

Subgroups

	subgroup	G/H
center	C₂	D₁₀₀₈₈
commutator, cosocle	C₈×C₁₃×C₉₇	C₂²
Frattini	C₈	C₂×D₁₂₆₁
Fitting	C₁₃×C₁₆×C₉₇	C₂
socle	C₂×C₁₃×C₉₇	D₈
hypercenter	C₁₆	D₁₂₆₁
hypocenter	C₁₃×C₉₇	D₁₆
LCS	G ▷ C₈×C₁₃×C₉₇ ▷ C₄×C₁₃×C₉₇ ▷ C₂×C₁₃×C₉₇ ▷ C₁₃×C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 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:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(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:

2^5,13,97|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^12,g^a=g^96

Conjugacy classes & character table

10091 conjugacy classes. The character table is too large to display.

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, 13, 97]);;
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.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^96);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D20176 #72274346