D₁₅₃₉₂ #72272152

order	30784
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₅₃₉₂

Splittings

Fitting	C₁₃×C₃₂×C₃₇ ⋊ C₂
Hall	C₁₃×C₃₇ ⋊ D₃₂
Sylow	C₃₇ ⋊ D₄₁₆
Sylow	C₁₃ ⋊ D₁₁₈₄

Subgroups

	subgroup	G/H
center	C₂	D₇₆₉₆
commutator, cosocle	C₁₃×C₁₆×C₃₇	C₂²
Frattini	C₁₆	C₂×D₄₈₁
Fitting	C₁₃×C₃₂×C₃₇	C₂
socle	C₂×C₁₃×C₃₇	D₁₆
hypercenter	C₃₂	D₄₈₁
hypocenter	C₁₃×C₃₇	D₃₂
LCS	G ▷ C₁₃×C₁₆×C₃₇ ▷ C₈×C₁₃×C₃₇ ▷ C₄×C₁₃×C₃₇ ▷ C₂×C₁₃×C₃₇ ▷ C₁₃×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 82
transitive degree	μ_t(G) = 15392

A minimal faithful permutation representation — 2 generators on 82 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);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)

Polycyclic presentation:

2^6,13,37|b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^12,h^a=h^36

Conjugacy classes & character table

7699 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₀₇₈₄
maximal normal subgroup of	D₃₀₇₈₄
maximal quotient of	Dic₁₅₃₉₂, D₃₀₇₈₄

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 13, 37]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
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, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^12);;
gap> SetConjugate(coll, 8, 1, F.8^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D15392 #72272152