D₁₆₆₈₈ #72272809

order	33376
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₆₆₈₈

Splittings

Fitting	C₇×C₁₆×C₁₄₉ ⋊ C₂
Hall	C₇×C₁₄₉ ⋊ D₁₆
Sylow	C₁₄₉ ⋊ D₁₁₂
Sylow	C₇ ⋊ D₂₃₈₄

Subgroups

	subgroup	G/H
center	C₂	D₈₃₄₄
commutator, cosocle	C₇×C₈×C₁₄₉	C₂²
Frattini	C₈	C₂×D₁₀₄₃
Fitting	C₇×C₁₆×C₁₄₉	C₂
socle	C₂×C₇×C₁₄₉	D₈
hypercenter	C₁₆	D₁₀₄₃
hypocenter	C₇×C₁₄₉	D₁₆
LCS	G ▷ C₇×C₈×C₁₄₉ ▷ C₄×C₇×C₁₄₉ ▷ C₂×C₇×C₁₄₉ ▷ C₇×C₁₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 172
transitive degree	μ_t(G) = 16688

A minimal faithful permutation representation — 2 generators on 172 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,23)(19,22)(20,21)(25,172)(26,171)(27,170)(28,169)(29,168)(30,167)(31,166)(32,165)(33,164)(34,163)(35,162)(36,161)(37,160)(38,159)(39,158)(40,157)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(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:

2^5,7,149|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^6,g^a=g^148

Conjugacy classes & character table

8347 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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 7, 149]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^148);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D16688 #72272809