D₁₉₃₁₇ #50311313

order	38634
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₉₃₁₇
unambiguous semidirect product	C₃⋊D₆₄₃₉
unambiguous semidirect product	C₄₇⋊D₄₁₁
unambiguous semidirect product	C₁₃₇⋊D₁₄₁

Splittings

Fitting, socle, derived, Hall	C₃×C₄₇×C₁₃₇ ⋊ C₂
Hall	C₄₇×C₁₃₇ ⋊ D₃
Hall	C₃×C₁₃₇ ⋊ D₄₇
Hall	C₃×C₄₇ ⋊ D₁₃₇
Sylow	C₁₃₇ ⋊ D₁₄₁
Sylow	C₄₇ ⋊ D₄₁₁
Sylow	C₃ ⋊ D₆₄₃₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₄₇×C₁₃₇	C₂
LCS	G ▷ C₃×C₄₇×C₁₃₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 187
transitive degree	μ_t(G) = 19317

A minimal faithful permutation representation — 2 generators on 187 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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187);(2,3)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(52,187)(53,186)(54,185)(55,184)(56,183)(57,182)(58,181)(59,180)(60,179)(61,178)(62,177)(63,176)(64,175)(65,174)(66,173)(67,172)(68,171)(69,170)(70,169)(71,168)(72,167)(73,166)(74,165)(75,164)(76,163)(77,162)(78,161)(79,160)(80,159)(81,158)(82,157)(83,156)(84,155)(85,154)(86,153)(87,152)(88,151)(89,150)(90,149)(91,148)(92,147)(93,146)(94,145)(95,144)(96,143)(97,142)(98,141)(99,140)(100,139)(101,138)(102,137)(103,136)(104,135)(105,134)(106,133)(107,132)(108,131)(109,130)(110,129)(111,128)(112,127)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,120)

Polycyclic presentation:

2,3,47,137|b^a=b^2,c^a=c^46,d^a=d^136

Conjugacy classes & character table

9660 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38634, 8);

D19317 #50311313