D₁₇₅₉₆ #72273205

order	35192
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₇₅₉₆

Splittings

Fitting	C₄×C₅₃×C₈₃ ⋊ C₂
Hall	C₅₃×C₈₃ ⋊ D₄
Sylow	C₈₃ ⋊ D₂₁₂
Sylow	C₅₃ ⋊ D₃₃₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₄₃₉₉
commutator, socle, cosocle	C₂×C₅₃×C₈₃	C₂²
Fitting	C₄×C₅₃×C₈₃	C₂
hypercenter	C₄	D₄₃₉₉
hypocenter	C₅₃×C₈₃	D₄
LCS	G ▷ C₂×C₅₃×C₈₃ ▷ C₅₃×C₈₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 140
transitive degree	μ_t(G) = 17596

A minimal faithful permutation representation — 2 generators on 140 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)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140);(2,4)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(59,140)(60,139)(61,138)(62,137)(63,136)(64,135)(65,134)(66,133)(67,132)(68,131)(69,130)(70,129)(71,128)(72,127)(73,126)(74,125)(75,124)(76,123)(77,122)(78,121)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)

Polycyclic presentation:

2^3,53,83|b^2=c,b^a=bc,d^a=d^52,e^a=e^82

Conjugacy classes & character table

8801 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 53, 83]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^52);;
gap> SetConjugate(coll, 5, 1, F.5^82);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D17596 #72273205