D₃₀₁₇₆ #72295482

order	60352
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₁₇₆

Splittings

Fitting	C₂₃×C₃₂×C₄₁ ⋊ C₂
Hall	C₂₃×C₄₁ ⋊ D₃₂
Sylow	C₄₁ ⋊ D₇₃₆
Sylow	C₂₃ ⋊ D₁₃₁₂

Subgroups

	subgroup	G/H
center	C₂	D₁₅₀₈₈
commutator, cosocle	C₁₆×C₂₃×C₄₁	C₂²
Frattini	C₁₆	C₂×D₉₄₃
Fitting	C₂₃×C₃₂×C₄₁	C₂
socle	C₂×C₂₃×C₄₁	D₁₆
hypercenter	C₃₂	D₉₄₃
hypocenter	C₂₃×C₄₁	D₃₂
LCS	G ▷ C₁₆×C₂₃×C₄₁ ▷ C₈×C₂₃×C₄₁ ▷ C₄×C₂₃×C₄₁ ▷ C₂×C₂₃×C₄₁ ▷ C₂₃×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 30176

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

Polycyclic presentation:

2^6,23,41|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^22,h^a=h^40

Conjugacy classes & character table

15091 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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 23, 41]);;
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.7^22);;
gap> SetConjugate(coll, 8, 1, F.8^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30176 #72295482