D₃₁₇₈₄ #72301744

order	63568
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₁₇₈₄

Splittings

Fitting	C₈×C₂₉×C₁₃₇ ⋊ C₂
Hall	C₂₉×C₁₃₇ ⋊ D₈
Sylow	C₁₃₇ ⋊ D₂₃₂
Sylow	C₂₉ ⋊ D₁₀₉₆

Subgroups

	subgroup	G/H
center	C₂	D₁₅₈₉₂
commutator, cosocle	C₄×C₂₉×C₁₃₇	C₂²
Frattini	C₄	C₂×D₃₉₇₃
Fitting	C₈×C₂₉×C₁₃₇	C₂
socle	C₂×C₂₉×C₁₃₇	D₄
hypercenter	C₈	D₃₉₇₃
hypocenter	C₂₉×C₁₃₇	D₈
LCS	G ▷ C₄×C₂₉×C₁₃₇ ▷ C₂×C₂₉×C₁₃₇ ▷ C₂₉×C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 174
transitive degree	μ_t(G) = 31784

A minimal faithful permutation representation — 2 generators on 174 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)(38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174);(2,8)(3,7)(4,6)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(39,174)(40,173)(41,172)(42,171)(43,170)(44,169)(45,168)(46,167)(47,166)(48,165)(49,164)(50,163)(51,162)(52,161)(53,160)(54,159)(55,158)(56,157)(57,156)(58,155)(59,154)(60,153)(61,152)(62,151)(63,150)(64,149)(65,148)(66,147)(67,146)(68,145)(69,144)(70,143)(71,142)(72,141)(73,140)(74,139)(75,138)(76,137)(77,136)(78,135)(79,134)(80,133)(81,132)(82,131)(83,130)(84,129)(85,128)(86,127)(87,126)(88,125)(89,124)(90,123)(91,122)(92,121)(93,120)(94,119)(95,118)(96,117)(97,116)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)

Polycyclic presentation:

2^4,29,137|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^28,f^a=f^136

Conjugacy classes & character table

15895 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, 29, 137]);;
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.5^28);;
gap> SetConjugate(coll, 6, 1, F.6^136);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D31784 #72301744