D₁₁₉₁₉ #37558621

order	23838
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₉₁₉
unambiguous semidirect product	C₃⋊D₃₉₇₃
unambiguous semidirect product	C₂₉⋊D₄₁₁
unambiguous semidirect product	C₁₃₇⋊D₈₇

Splittings

Fitting, socle, derived, Hall	C₃×C₂₉×C₁₃₇ ⋊ C₂
Hall	C₂₉×C₁₃₇ ⋊ D₃
Hall	C₃×C₁₃₇ ⋊ D₂₉
Sylow	C₁₃₇ ⋊ D₈₇
Hall	C₃×C₂₉ ⋊ D₁₃₇
Sylow	C₂₉ ⋊ D₄₁₁
Sylow	C₃ ⋊ D₃₉₇₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₂₉×C₁₃₇	C₂
LCS	G ▷ C₃×C₂₉×C₁₃₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 169
transitive degree	μ_t(G) = 11919

A minimal faithful permutation representation — 2 generators on 169 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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169);(2,3)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(34,169)(35,168)(36,167)(37,166)(38,165)(39,164)(40,163)(41,162)(42,161)(43,160)(44,159)(45,158)(46,157)(47,156)(48,155)(49,154)(50,153)(51,152)(52,151)(53,150)(54,149)(55,148)(56,147)(57,146)(58,145)(59,144)(60,143)(61,142)(62,141)(63,140)(64,139)(65,138)(66,137)(67,136)(68,135)(69,134)(70,133)(71,132)(72,131)(73,130)(74,129)(75,128)(76,127)(77,126)(78,125)(79,124)(80,123)(81,122)(82,121)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)

Polycyclic presentation:

2,3,29,137|b^a=b^2,c^a=c^28,d^a=d^136

Conjugacy classes & character table

5961 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23838, 8);

D11919 #37558621