D₂₈₄₉₆ #72292123

order	56992
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₈₄₉₆

Splittings

Fitting	C₁₃×C₁₆×C₁₃₇ ⋊ C₂
Hall	C₁₃×C₁₃₇ ⋊ D₁₆
Sylow	C₁₃₇ ⋊ D₂₀₈
Sylow	C₁₃ ⋊ D₂₁₉₂

Subgroups

	subgroup	G/H
center	C₂	D₁₄₂₄₈
commutator, cosocle	C₈×C₁₃×C₁₃₇	C₂²
Frattini	C₈	C₂×D₁₇₈₁
Fitting	C₁₃×C₁₆×C₁₃₇	C₂
socle	C₂×C₁₃×C₁₃₇	D₈
hypercenter	C₁₆	D₁₇₈₁
hypocenter	C₁₃×C₁₃₇	D₁₆
LCS	G ▷ C₈×C₁₃×C₁₃₇ ▷ C₄×C₁₃×C₁₃₇ ▷ C₂×C₁₃×C₁₃₇ ▷ C₁₃×C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 166
transitive degree	μ_t(G) = 28496

A minimal faithful permutation representation — 2 generators on 166 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(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:

2^5,13,137|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^12,g^a=g^136

Conjugacy classes & character table

14251 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, 13, 137]);;
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.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^136);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D28496 #72292123