D₁₃₅₂ #72210172

order	2704
automorphism	Aut(G) order 843648
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₅₂

Splittings

Fitting	C₈×C₁₆₉ ⋊ C₂
Sylow	C₁₆₉ ⋊ D₈

Subgroups

	subgroup	G/H
center	C₂	D₆₇₆
commutator, cosocle	C₄×C₁₆₉	C₂²
Frattini	C₄×C₁₃	C₂×D₁₃
Fitting	C₈×C₁₆₉	C₂
socle	C₂×C₁₃	D₅₂
hypercenter	C₈	D₁₆₉
hypocenter	C₁₆₉	D₈
LCS	G ▷ C₄×C₁₆₉ ▷ C₂×C₁₆₉ ▷ C₁₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

33 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 177
transitive degree	μ_t(G) = 1352

A minimal faithful permutation representation — 2 generators on 177 points:

(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177);(2,8)(3,7)(4,6)(10,177)(11,176)(12,175)(13,174)(14,173)(15,172)(16,171)(17,170)(18,169)(19,168)(20,167)(21,166)(22,165)(23,164)(24,163)(25,162)(26,161)(27,160)(28,159)(29,158)(30,157)(31,156)(32,155)(33,154)(34,153)(35,152)(36,151)(37,150)(38,149)(39,148)(40,147)(41,146)(42,145)(43,144)(44,143)(45,142)(46,141)(47,140)(48,139)(49,138)(50,137)(51,136)(52,135)(53,134)(54,133)(55,132)(56,131)(57,130)(58,129)(59,128)(60,127)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,120)(68,119)(69,118)(70,117)(71,116)(72,115)(73,114)(74,113)(75,112)(76,111)(77,110)(78,109)(79,108)(80,107)(81,106)(82,105)(83,104)(84,103)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)

Polycyclic presentation:

2^4,13^2|b^2=cd,c^2=d,e^13=f,b^a=bc,c^a=cd,e^a=e^12f^12,f^a=f^12

Conjugacy classes & character table

679 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₇₀₄, D₄₀₅₆, D₆₇₆₀, …
maximal normal subgroup of	D₂₇₀₄
maximal quotient of	D₂₇₀₄, D₄₀₅₆, Dic₁₃₅₂, …

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, 13, 13]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^12*F.6^12);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1352 #72210172