D₁₉₇₈₈ #72274155

order	39576
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₉₇₈₈

Splittings

Fitting	C₃×C₄×C₁₇×C₉₇ ⋊ C₂
Hall	C₃×C₁₇×C₉₇ ⋊ D₄
Hall	C₁₇×C₉₇ ⋊ D₁₂
Hall	C₃×C₉₇ ⋊ D₆₈
Sylow	C₉₇ ⋊ D₂₀₄
Hall	C₃×C₁₇ ⋊ D₃₈₈
Sylow	C₁₇ ⋊ D₁₁₆₄
Sylow	C₃ ⋊ D₆₅₉₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₄₉₄₇
commutator, socle, cosocle	C₂×C₃×C₁₇×C₉₇	C₂²
Fitting	C₃×C₄×C₁₇×C₉₇	C₂
hypercenter	C₄	D₄₉₄₇
hypocenter	C₃×C₁₇×C₉₇	D₄
LCS	G ▷ C₂×C₃×C₁₇×C₉₇ ▷ C₃×C₁₇×C₉₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 19788

A minimal faithful permutation representation — 2 generators on 121 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);(2,4)(6,7)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(26,121)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)

Polycyclic presentation:

2^3,3,17,97|b^2=c,b^a=bc,d^a=d^2,e^a=e^16,f^a=f^96

Conjugacy classes & character table

9897 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 17, 97]);;
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.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^96);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D19788 #72274155