D₂₀₇₉₂ #72274693

order	41584
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₀₇₉₂

Splittings

Fitting	C₈×C₂₃×C₁₁₃ ⋊ C₂
Hall	C₂₃×C₁₁₃ ⋊ D₈
Sylow	C₁₁₃ ⋊ D₁₈₄
Sylow	C₂₃ ⋊ D₉₀₄

Subgroups

	subgroup	G/H
center	C₂	D₁₀₃₉₆
commutator, cosocle	C₄×C₂₃×C₁₁₃	C₂²
Frattini	C₄	C₂×D₂₅₉₉
Fitting	C₈×C₂₃×C₁₁₃	C₂
socle	C₂×C₂₃×C₁₁₃	D₄
hypercenter	C₈	D₂₅₉₉
hypocenter	C₂₃×C₁₁₃	D₈
LCS	G ▷ C₄×C₂₃×C₁₁₃ ▷ C₂×C₂₃×C₁₁₃ ▷ C₂₃×C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 144
transitive degree	μ_t(G) = 20792

A minimal faithful permutation representation — 2 generators on 144 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)(32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144);(2,8)(3,7)(4,6)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(33,144)(34,143)(35,142)(36,141)(37,140)(38,139)(39,138)(40,137)(41,136)(42,135)(43,134)(44,133)(45,132)(46,131)(47,130)(48,129)(49,128)(50,127)(51,126)(52,125)(53,124)(54,123)(55,122)(56,121)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)

Polycyclic presentation:

2^4,23,113|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^22,f^a=f^112

Conjugacy classes & character table

10399 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, 2, 23, 113]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^112);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D20792 #72274693