D₁₆₆₉₂ #72272811

order	33384
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₆₆₉₂

Splittings

Fitting	C₃×C₄×C₁₃×C₁₀₇ ⋊ C₂
Hall	C₃×C₁₃×C₁₀₇ ⋊ D₄
Hall	C₁₃×C₁₀₇ ⋊ D₁₂
Hall	C₃×C₁₀₇ ⋊ D₅₂
Sylow	C₁₀₇ ⋊ D₁₅₆
Hall	C₃×C₁₃ ⋊ D₄₂₈
Sylow	C₁₃ ⋊ D₁₂₈₄
Sylow	C₃ ⋊ D₅₅₆₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₄₁₇₃
commutator, socle, cosocle	C₂×C₃×C₁₃×C₁₀₇	C₂²
Fitting	C₃×C₄×C₁₃×C₁₀₇	C₂
hypercenter	C₄	D₄₁₇₃
hypocenter	C₃×C₁₃×C₁₀₇	D₄
LCS	G ▷ C₂×C₃×C₁₃×C₁₀₇ ▷ C₃×C₁₃×C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 127
transitive degree	μ_t(G) = 16692

A minimal faithful permutation representation — 2 generators on 127 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);(2,4)(6,7)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(22,127)(23,126)(24,125)(25,124)(26,123)(27,122)(28,121)(29,120)(30,119)(31,118)(32,117)(33,116)(34,115)(35,114)(36,113)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)

Polycyclic presentation:

2^3,3,13,107|b^2=c,b^a=bc,d^a=d^2,e^a=e^12,f^a=f^106

Conjugacy classes & character table

8349 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 13, 107]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^106);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D16692 #72272811