D₁₇₄₄₄ #72273142

order	34888
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₇₄₄₄

Splittings

Fitting	C₄×C₄₉×C₈₉ ⋊ C₂
Hall	C₄₉×C₈₉ ⋊ D₄
Sylow	C₈₉ ⋊ D₁₉₆
Sylow	C₄₉ ⋊ D₃₅₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄₃₆₁
commutator, cosocle	C₂×C₄₉×C₈₉	C₂²
Frattini	C₂×C₇	C₂×D₆₂₃
Fitting	C₄×C₄₉×C₈₉	C₂
socle	C₂×C₇×C₈₉	C₂×D₇
hypercenter	C₄	D₄₃₆₁
hypocenter	C₄₉×C₈₉	D₄
LCS	G ▷ C₂×C₄₉×C₈₉ ▷ C₄₉×C₈₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 142
transitive degree	μ_t(G) = 17444

A minimal faithful permutation representation — 2 generators on 142 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)(54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142);(2,4)(6,53)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(55,142)(56,141)(57,140)(58,139)(59,138)(60,137)(61,136)(62,135)(63,134)(64,133)(65,132)(66,131)(67,130)(68,129)(69,128)(70,127)(71,126)(72,125)(73,124)(74,123)(75,122)(76,121)(77,120)(78,119)(79,118)(80,117)(81,116)(82,115)(83,114)(84,113)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)

Polycyclic presentation:

2^3,7,89,7|b^2=c,d^7=f,b^a=bc,d^a=d^6f^6,e^a=e^88,f^a=f^6

Conjugacy classes & character table

8725 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, 7, 89, 7]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6*F.6^6);;
gap> SetConjugate(coll, 5, 1, F.5^88);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D17444 #72273142