D₁₉₉₀₈ #72274205

order	39816
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₉₉₀₈

Splittings

Fitting	C₄×C₇×C₉×C₇₉ ⋊ C₂
Hall	C₇×C₉×C₇₉ ⋊ D₄
Hall	C₉×C₇₉ ⋊ D₂₈
Hall	C₇×C₇₉ ⋊ D₃₆
Sylow	C₇₉ ⋊ D₂₅₂
Hall	C₇×C₉ ⋊ D₃₁₆
Sylow	C₉ ⋊ D₂₂₁₂
Sylow	C₇ ⋊ D₂₈₄₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄₉₇₇
commutator, cosocle	C₂×C₇×C₉×C₇₉	C₂²
Frattini	C₂×C₃	C₂×D₁₆₅₉
Fitting	C₄×C₇×C₉×C₇₉	C₂
socle	C₂×C₃×C₇×C₇₉	C₂×D₃
hypercenter	C₄	D₄₉₇₇
hypocenter	C₇×C₉×C₇₉	D₄
LCS	G ▷ C₂×C₇×C₉×C₇₉ ▷ C₇×C₉×C₇₉
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 19908

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

Polycyclic presentation:

2^3,3,7,79,3|b^2=c,d^3=g,b^a=bc,d^a=d^2g^2,e^a=e^6,f^a=f^78,g^a=g^2

Conjugacy classes & character table

9957 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, 3, 7, 79, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^78);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D19908 #72274205