D₃₁₆₂₄ #72301544

order	63248
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₁₆₂₄

Splittings

Fitting	C₈×C₅₉×C₆₇ ⋊ C₂
Hall	C₅₉×C₆₇ ⋊ D₈
Sylow	C₆₇ ⋊ D₄₇₂
Sylow	C₅₉ ⋊ D₅₃₆

Subgroups

	subgroup	G/H
center	C₂	D₁₅₈₁₂
commutator, cosocle	C₄×C₅₉×C₆₇	C₂²
Frattini	C₄	C₂×D₃₉₅₃
Fitting	C₈×C₅₉×C₆₇	C₂
socle	C₂×C₅₉×C₆₇	D₄
hypercenter	C₈	D₃₉₅₃
hypocenter	C₅₉×C₆₇	D₈
LCS	G ▷ C₄×C₅₉×C₆₇ ▷ C₂×C₅₉×C₆₇ ▷ C₅₉×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 134
transitive degree	μ_t(G) = 31624

A minimal faithful permutation representation — 2 generators on 134 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);(2,8)(3,7)(4,6)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(69,134)(70,133)(71,132)(72,131)(73,130)(74,129)(75,128)(76,127)(77,126)(78,125)(79,124)(80,123)(81,122)(82,121)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)

Polycyclic presentation:

2^4,59,67|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^58,f^a=f^66

Conjugacy classes & character table

15815 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, 59, 67]);;
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.5^58);;
gap> SetConjugate(coll, 6, 1, F.6^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D31624 #72301544