D₃×#19129429 #45654265

order	32526
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×#19129429
unambiguous semidirect product	C₃⋊(C₂×#19129429)

Splittings

Fitting, socle, derived	C₃×C₁₃×C₁₃₉ ⋊ C₂×C₃
Hall	C₃×#19129429 ⋊ C₂
Hall	C₁₃×C₁₃₉ ⋊ C₃×D₃
Sylow	C₁₃₉ ⋊ D₃×C₁₃⋊C₃
Sylow	C₁₃ ⋊ D₃×C₁₃₉⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₃×C₁₃₉	C₂×C₃
LCS	G ▷ C₃×C₁₃×C₁₃₉
UCS	C₁

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 155
transitive degree	μ_t(G) = 5421

A minimal faithful permutation representation — 2 generators on 155 points:

(2,3)(5,6,8)(7,10,14)(9,12,11)(13,15,16)(18,19,21)(20,23,28)(22,26,25)(24,30,38)(27,33,35)(29,36,32)(31,40,51)(34,43,48)(37,46,45)(39,49,42)(41,53,67)(44,56,64)(47,59,61)(50,62,58)(52,65,55)(54,69,86)(57,72,83)(60,75,80)(63,78,77)(66,81,74)(68,84,71)(70,88,107)(73,91,104)(76,94,102)(79,97,99)(82,100,96)(85,89,93)(87,105,90)(92,110,125)(95,113,123)(98,116,121)(101,119,118)(103,111,115)(106,124,112)(108,126,109)(114,129,139)(117,131,138)(120,134,136)(122,130,133)(127,140,128)(132,141,148)(135,143,147)(137,142,145)(144,149,153)(146,150,152)(151,154,155);(1,2,3)(4,5,7,8,11,10,14,16,15,6,9,13,12)(17,18,20,24,31,41,54,70,89,65,38,48,61,77,96,115,88,108,127,126,86,104,123,138,147,152,141,113,72,40,52,66,82,101,120,135,144,150,134,97,59,33,19,22,27,34,44,57,73,92,111,81,49,28,35,45,58,74,93,112,128,140,105,67,83,102,121,136,145,129,91,53,68,85,103,122,137,146,151,154,143,116,75,43,23,29,37,47,60,76,95,114,130,100,62,36,21,25,32,42,55,71,90,109,124,84,51,64,80,99,118,133,110,69,87,106,107,125,139,148,153,155,149,131,94,56,30,39,50,63,79,98,117,132,142,119,78,46,26)

Conjugacy classes & character table

1815 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32526, 54);

D3×#19129429 #45654265