D₂₇₃₈₄ #72290692

order	54768
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₃₈₄

Splittings

Fitting	C₃×C₇×C₈×C₁₆₃ ⋊ C₂
Hall	C₃×C₇×C₁₆₃ ⋊ D₈
Hall	C₇×C₁₆₃ ⋊ D₂₄
Hall	C₃×C₁₆₃ ⋊ D₅₆
Sylow	C₁₆₃ ⋊ D₁₆₈
Hall	C₃×C₇ ⋊ D₁₃₀₄
Sylow	C₇ ⋊ D₃₉₁₂
Sylow	C₃ ⋊ D₉₁₂₈

Subgroups

	subgroup	G/H
center	C₂	D₁₃₆₉₂
commutator, cosocle	C₃×C₄×C₇×C₁₆₃	C₂²
Frattini	C₄	C₂×D₃₄₂₃
Fitting	C₃×C₇×C₈×C₁₆₃	C₂
socle	C₂×C₃×C₇×C₁₆₃	D₄
hypercenter	C₈	D₃₄₂₃
hypocenter	C₃×C₇×C₁₆₃	D₈
LCS	G ▷ C₃×C₄×C₇×C₁₆₃ ▷ C₂×C₃×C₇×C₁₆₃ ▷ C₃×C₇×C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 181
transitive degree	μ_t(G) = 27384

A minimal faithful permutation representation — 2 generators on 181 points:

(1,2,3,4,5,6,7,8)(9,10,11)(12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181);(2,8)(3,7)(4,6)(10,11)(13,18)(14,17)(15,16)(20,181)(21,180)(22,179)(23,178)(24,177)(25,176)(26,175)(27,174)(28,173)(29,172)(30,171)(31,170)(32,169)(33,168)(34,167)(35,166)(36,165)(37,164)(38,163)(39,162)(40,161)(41,160)(42,159)(43,158)(44,157)(45,156)(46,155)(47,154)(48,153)(49,152)(50,151)(51,150)(52,149)(53,148)(54,147)(55,146)(56,145)(57,144)(58,143)(59,142)(60,141)(61,140)(62,139)(63,138)(64,137)(65,136)(66,135)(67,134)(68,133)(69,132)(70,131)(71,130)(72,129)(73,128)(74,127)(75,126)(76,125)(77,124)(78,123)(79,122)(80,121)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)(91,110)(92,109)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101)

Polycyclic presentation:

2^4,3,7,163|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^6,g^a=g^162

Conjugacy classes & character table

13695 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, 3, 7, 163]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^162);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27384 #72290692