D₂₆₁₃₆ #72289389

order	52272
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₁₃₆

Splittings

Fitting	C₈×C₂₇×C₁₂₁ ⋊ C₂
Hall	C₂₇×C₁₂₁ ⋊ D₈
Sylow	C₁₂₁ ⋊ D₂₁₆
Sylow	C₂₇ ⋊ D₉₆₈

Subgroups

	subgroup	G/H
center	C₂	D₁₃₀₆₈
commutator, cosocle	C₄×C₂₇×C₁₂₁	C₂²
Frattini	C₄×C₉×C₁₁	C₂×D₃₃
Fitting	C₈×C₂₇×C₁₂₁	C₂
socle	C₂×C₃×C₁₁	D₃₉₆
hypercenter	C₈	D₃₂₆₇
hypocenter	C₂₇×C₁₂₁	D₈
LCS	G ▷ C₄×C₂₇×C₁₂₁ ▷ C₂×C₂₇×C₁₂₁ ▷ C₂₇×C₁₂₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 156
transitive degree	μ_t(G) = 26136

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

Polycyclic presentation:

2^4,3,11,3,11,3|b^2=cd,c^2=d,e^3=g,f^11=h,g^3=i,b^a=bc,c^a=cd,e^a=e^2g^2i^2,f^a=f^10h^10,g^a=g^2i^2,h^a=h^10,i^a=i^2

Conjugacy classes & character table

13071 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", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 11, 3, 11, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2*F.9^2);;
gap> SetConjugate(coll, 6, 1, F.6^10*F.8^10);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^10);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26136 #72289389