D₂₉₈₇₆ #72294100

order	59752
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₉₈₇₆

Splittings

Fitting	C₄×C₇×C₁₁×C₉₇ ⋊ C₂
Hall	C₇×C₁₁×C₉₇ ⋊ D₄
Hall	C₁₁×C₉₇ ⋊ D₂₈
Hall	C₇×C₉₇ ⋊ D₄₄
Sylow	C₉₇ ⋊ D₃₀₈
Hall	C₇×C₁₁ ⋊ D₃₈₈
Sylow	C₁₁ ⋊ D₂₇₁₆
Sylow	C₇ ⋊ D₄₂₆₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₄₆₉
commutator, socle, cosocle	C₂×C₇×C₁₁×C₉₇	C₂²
Fitting	C₄×C₇×C₁₁×C₉₇	C₂
hypercenter	C₄	D₇₄₆₉
hypocenter	C₇×C₁₁×C₉₇	D₄
LCS	G ▷ C₂×C₇×C₁₁×C₉₇ ▷ C₇×C₁₁×C₉₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 29876

A minimal faithful permutation representation — 2 generators on 119 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);(2,4)(6,11)(7,10)(8,9)(13,22)(14,21)(15,20)(16,19)(17,18)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2^3,7,11,97|b^2=c,b^a=bc,d^a=d^6,e^a=e^10,f^a=f^96

Conjugacy classes & character table

14941 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, 7, 11, 97]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^96);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D29876 #72294100