D₁₃₉₃₆ #72235626

order	27872
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₉₃₆

Splittings

Fitting	C₁₃×C₁₆×C₆₇ ⋊ C₂
Hall	C₁₃×C₆₇ ⋊ D₁₆
Sylow	C₆₇ ⋊ D₂₀₈
Sylow	C₁₃ ⋊ D₁₀₇₂

Subgroups

	subgroup	G/H
center	C₂	D₆₉₆₈
commutator, cosocle	C₈×C₁₃×C₆₇	C₂²
Frattini	C₈	C₂×D₈₇₁
Fitting	C₁₃×C₁₆×C₆₇	C₂
socle	C₂×C₁₃×C₆₇	D₈
hypercenter	C₁₆	D₈₇₁
hypocenter	C₁₃×C₆₇	D₁₆
LCS	G ▷ C₈×C₁₃×C₆₇ ▷ C₄×C₁₃×C₆₇ ▷ C₂×C₁₃×C₆₇ ▷ C₁₃×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 13936

A minimal faithful permutation representation — 2 generators on 96 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);(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)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Polycyclic presentation:

2^5,13,67|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^66

Conjugacy classes & character table

6971 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₇₈₇₂
maximal normal subgroup of	D₂₇₈₇₂
maximal quotient of	Dic₁₃₉₃₆, D₂₇₈₇₂

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, 67]);;
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.7^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D13936 #72235626