D₃₀₁₀₄ #72295425

order	60208
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₁₀₄

Splittings

Fitting	C₈×C₅₃×C₇₁ ⋊ C₂
Hall	C₅₃×C₇₁ ⋊ D₈
Sylow	C₇₁ ⋊ D₄₂₄
Sylow	C₅₃ ⋊ D₅₆₈

Subgroups

	subgroup	G/H
center	C₂	D₁₅₀₅₂
commutator, cosocle	C₄×C₅₃×C₇₁	C₂²
Frattini	C₄	C₂×D₃₇₆₃
Fitting	C₈×C₅₃×C₇₁	C₂
socle	C₂×C₅₃×C₇₁	D₄
hypercenter	C₈	D₃₇₆₃
hypocenter	C₅₃×C₇₁	D₈
LCS	G ▷ C₄×C₅₃×C₇₁ ▷ C₂×C₅₃×C₇₁ ▷ C₅₃×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 30104

A minimal faithful permutation representation — 2 generators on 132 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);(2,8)(3,7)(4,6)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(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:

2^4,53,71|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^52,f^a=f^70

Conjugacy classes & character table

15055 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, 53, 71]);;
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.5^52);;
gap> SetConjugate(coll, 6, 1, F.6^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30104 #72295425