D₁₀₃₈₀ #72215333

order	20760
automorphism	Aut(G) order 28565760
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₀₃₈₀

Splittings

Fitting	C₃×C₄×C₅×C₁₇₃ ⋊ C₂
Hall	C₃×C₅×C₁₇₃ ⋊ D₄
Hall	C₅×C₁₇₃ ⋊ D₁₂
Hall	C₃×C₁₇₃ ⋊ D₂₀
Sylow	C₁₇₃ ⋊ D₆₀
Hall	C₃×C₅ ⋊ D₆₉₂
Sylow	C₅ ⋊ D₂₀₇₆
Sylow	C₃ ⋊ D₃₄₆₀

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₅₉₅
commutator, socle, cosocle	C₂×C₃×C₅×C₁₇₃	C₂²
Fitting	C₃×C₄×C₅×C₁₇₃	C₂
hypercenter	C₄	D₂₅₉₅
hypocenter	C₃×C₅×C₁₇₃	D₄
LCS	G ▷ C₂×C₃×C₅×C₁₇₃ ▷ C₃×C₅×C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 185
transitive degree	μ_t(G) = 10380

A minimal faithful permutation representation — 2 generators on 185 points:

(1,2,3,4)(5,6,7)(8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185);(2,4)(6,7)(9,12)(10,11)(14,185)(15,184)(16,183)(17,182)(18,181)(19,180)(20,179)(21,178)(22,177)(23,176)(24,175)(25,174)(26,173)(27,172)(28,171)(29,170)(30,169)(31,168)(32,167)(33,166)(34,165)(35,164)(36,163)(37,162)(38,161)(39,160)(40,159)(41,158)(42,157)(43,156)(44,155)(45,154)(46,153)(47,152)(48,151)(49,150)(50,149)(51,148)(52,147)(53,146)(54,145)(55,144)(56,143)(57,142)(58,141)(59,140)(60,139)(61,138)(62,137)(63,136)(64,135)(65,134)(66,133)(67,132)(68,131)(69,130)(70,129)(71,128)(72,127)(73,126)(74,125)(75,124)(76,123)(77,122)(78,121)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)

Polycyclic presentation:

2^3,3,5,173|b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^172

Conjugacy classes & character table

5193 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₀₇₆₀, D₃₁₁₄₀
maximal normal subgroup of	D₂₀₇₆₀
maximal quotient of	Dic₁₀₃₈₀, D₂₀₇₆₀, D₃₁₁₄₀

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, 3, 5, 173]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^172);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D10380 #72215333