D₂₈₅₅₂ #72292173

order	57104
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₈₅₅₂

Splittings

Fitting	C₈×C₄₃×C₈₃ ⋊ C₂
Hall	C₄₃×C₈₃ ⋊ D₈
Sylow	C₈₃ ⋊ D₃₄₄
Sylow	C₄₃ ⋊ D₆₆₄

Subgroups

	subgroup	G/H
center	C₂	D₁₄₂₇₆
commutator, cosocle	C₄×C₄₃×C₈₃	C₂²
Frattini	C₄	C₂×D₃₅₆₉
Fitting	C₈×C₄₃×C₈₃	C₂
socle	C₂×C₄₃×C₈₃	D₄
hypercenter	C₈	D₃₅₆₉
hypocenter	C₄₃×C₈₃	D₈
LCS	G ▷ C₄×C₄₃×C₈₃ ▷ C₂×C₄₃×C₈₃ ▷ C₄₃×C₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 134
transitive degree	μ_t(G) = 28552

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)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(53,134)(54,133)(55,132)(56,131)(57,130)(58,129)(59,128)(60,127)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,120)(68,119)(69,118)(70,117)(71,116)(72,115)(73,114)(74,113)(75,112)(76,111)(77,110)(78,109)(79,108)(80,107)(81,106)(82,105)(83,104)(84,103)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)

Polycyclic presentation:

2^4,43,83|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^42,f^a=f^82

Conjugacy classes & character table

14279 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, 43, 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.5^42);;
gap> SetConjugate(coll, 6, 1, F.6^82);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D28552 #72292173