D₃₉₂₄ #72211263

order	7848
automorphism	Aut(G) order 5085504
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₉₂₄

Splittings

Fitting	C₄×C₉×C₁₀₉ ⋊ C₂
Hall	C₉×C₁₀₉ ⋊ D₄
Sylow	C₁₀₉ ⋊ D₃₆
Sylow	C₉ ⋊ D₄₃₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₉₈₁
commutator, cosocle	C₂×C₉×C₁₀₉	C₂²
Frattini	C₂×C₃	C₂×D₃₂₇
Fitting	C₄×C₉×C₁₀₉	C₂
socle	C₂×C₃×C₁₀₉	C₂×D₃
hypercenter	C₄	D₉₈₁
hypocenter	C₉×C₁₀₉	D₄
LCS	G ▷ C₂×C₉×C₁₀₉ ▷ C₉×C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 122
transitive degree	μ_t(G) = 3924

A minimal faithful permutation representation — 2 generators on 122 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);(2,4)(6,13)(7,12)(8,11)(9,10)(15,122)(16,121)(17,120)(18,119)(19,118)(20,117)(21,116)(22,115)(23,114)(24,113)(25,112)(26,111)(27,110)(28,109)(29,108)(30,107)(31,106)(32,105)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Polycyclic presentation:

2^3,3,109,3|b^2=c,d^3=f,b^a=bc,d^a=d^2f^2,e^a=e^108,f^a=f^2

Conjugacy classes & character table

1965 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇₈₄₈, D₁₁₇₇₂, D₁₉₆₂₀, …
maximal normal subgroup of	D₇₈₄₈
maximal quotient of	D₇₈₄₈, Dic₃₉₂₄, D₁₁₇₇₂, …

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, 109, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2);;
gap> SetConjugate(coll, 5, 1, F.5^108);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D3924 #72211263