D₁₈₃₉₂ #72273545

order	36784
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₈₃₉₂

Splittings

Fitting	C₈×C₁₉×C₁₂₁ ⋊ C₂
Hall	C₁₉×C₁₂₁ ⋊ D₈
Sylow	C₁₂₁ ⋊ D₁₅₂
Sylow	C₁₉ ⋊ D₉₆₈

Subgroups

	subgroup	G/H
center	C₂	D₉₁₉₆
commutator, cosocle	C₄×C₁₉×C₁₂₁	C₂²
Frattini	C₄×C₁₁	C₂×D₂₀₉
Fitting	C₈×C₁₉×C₁₂₁	C₂
socle	C₂×C₁₁×C₁₉	D₄₄
hypercenter	C₈	D₂₂₉₉
hypocenter	C₁₉×C₁₂₁	D₈
LCS	G ▷ C₄×C₁₉×C₁₂₁ ▷ C₂×C₁₉×C₁₂₁ ▷ C₁₉×C₁₂₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 148
transitive degree	μ_t(G) = 18392

A minimal faithful permutation representation — 2 generators on 148 points:

(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129)(130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148);(2,8)(3,7)(4,6)(10,129)(11,128)(12,127)(13,126)(14,125)(15,124)(16,123)(17,122)(18,121)(19,120)(20,119)(21,118)(22,117)(23,116)(24,115)(25,114)(26,113)(27,112)(28,111)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(131,148)(132,147)(133,146)(134,145)(135,144)(136,143)(137,142)(138,141)(139,140)

Polycyclic presentation:

2^4,11,19,11|b^2=cd,c^2=d,e^11=g,b^a=bc,c^a=cd,e^a=e^10g^10,f^a=f^18,g^a=g^10

Conjugacy classes & character table

9199 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, 11, 19, 11]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^10*F.7^10);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D18392 #72273545