D₃₁₂₀₈ #72300285

order	62416
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₁₂₀₈

Splittings

Fitting	C₈×C₄₇×C₈₃ ⋊ C₂
Hall	C₄₇×C₈₃ ⋊ D₈
Sylow	C₈₃ ⋊ D₃₇₆
Sylow	C₄₇ ⋊ D₆₆₄

Subgroups

	subgroup	G/H
center	C₂	D₁₅₆₀₄
commutator, cosocle	C₄×C₄₇×C₈₃	C₂²
Frattini	C₄	C₂×D₃₉₀₁
Fitting	C₈×C₄₇×C₈₃	C₂
socle	C₂×C₄₇×C₈₃	D₄
hypercenter	C₈	D₃₉₀₁
hypocenter	C₄₇×C₈₃	D₈
LCS	G ▷ C₄×C₄₇×C₈₃ ▷ C₂×C₄₇×C₈₃ ▷ C₄₇×C₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 138
transitive degree	μ_t(G) = 31208

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

Polycyclic presentation:

2^4,47,83|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^46,f^a=f^82

Conjugacy classes & character table

15607 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, 2, 47, 83]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^46);;
gap> SetConjugate(coll, 6, 1, F.6^82);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D31208 #72300285