D₁₉₃₂₃ #50311856

order	38646
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₉₃₂₃
unambiguous semidirect product	C₉⋊D₂₁₄₇
unambiguous semidirect product	C₁₉⋊D₁₀₁₇
unambiguous semidirect product	C₁₁₃⋊D₁₇₁

Splittings

Fitting, derived, Hall	C₉×C₁₉×C₁₁₃ ⋊ C₂
Hall	C₁₉×C₁₁₃ ⋊ D₉
Hall	C₉×C₁₁₃ ⋊ D₁₉
Hall	C₉×C₁₉ ⋊ D₁₁₃
Sylow	C₁₁₃ ⋊ D₁₇₁
Sylow	C₁₉ ⋊ D₁₀₁₇
Sylow	C₉ ⋊ D₂₁₄₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₉×C₁₁₃	C₂
Frattini	C₃	D₆₄₄₁
socle	C₃×C₁₉×C₁₁₃	D₃
LCS	G ▷ C₉×C₁₉×C₁₁₃
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 141
transitive degree	μ_t(G) = 19323

A minimal faithful permutation representation — 2 generators on 141 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);(2,9)(3,8)(4,7)(5,6)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(30,141)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,134)(38,133)(39,132)(40,131)(41,130)(42,129)(43,128)(44,127)(45,126)(46,125)(47,124)(48,123)(49,122)(50,121)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)

Polycyclic presentation:

2,3,19,113,3|b^3=e^2,b^a=b^2e,c^a=c^18,d^a=d^112,e^a=e^2

Conjugacy classes & character table

9663 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38646, 9);

D19323 #50311856