D₃₉₆₉ #72211265

order	7938
automorphism	Aut(G) order 9001692
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₉₆₉

Splittings

Fitting, derived, Hall	C₄₉×C₈₁ ⋊ C₂
Sylow	C₈₁ ⋊ D₄₉
Sylow	C₄₉ ⋊ D₈₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₄₉×C₈₁	C₂
Frattini	C₇×C₂₇	D₂₁
socle	C₃×C₇	D₁₈₉
LCS	G ▷ C₄₉×C₈₁
UCS	C₁

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 130
transitive degree	μ_t(G) = 3969

A minimal faithful permutation representation — 2 generators on 130 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);(2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(83,130)(84,129)(85,128)(86,127)(87,126)(88,125)(89,124)(90,123)(91,122)(92,121)(93,120)(94,119)(95,118)(96,117)(97,116)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)

Polycyclic presentation:

2,3,7,3,7,3^2|b^3=d,c^7=e,d^3=f,f^3=g,b^a=b^2d^2f^2g^2,c^a=c^6e^6,d^a=d^2f^2g^2,e^a=e^6,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

1986 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₃₉₆₉, D₁₁₉₀₇, D₁₉₈₄₅, …
maximal normal subgroup of	C₂×D₃₉₆₉
maximal quotient of	C₂×D₃₉₆₉, Dic₃₉₆₉, D₁₁₉₀₇, …

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, 3, 7, 3, 7, 3, 3]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.4^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 3, 1, F.3^6*F.5^6);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D3969 #72211265