D₃×#19128629 #44544826

order	32004
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

D₃×#19128629

Splittings

Fitting, socle, derived	C₃×C₇×C₁₂₇ ⋊ C₂²×C₃
Hall	C₃×#14581887 ⋊ C₂²
Hall	C₇×C₁₂₇ ⋊ C₂×C₃×D₃
Sylow	C₁₂₇ ⋊ D₃×AGL(1,𝔽₇)
Sylow	C₇ ⋊ C₂×D₃×C₁₂₇⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₁₂₇	C₂²×C₃
LCS	G ▷ C₃×C₇×C₁₂₇
UCS	C₁

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 137
transitive degree	μ_t(G) = 2667

A minimal faithful permutation representation — 2 generators on 137 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32004, 208);

D3×#19128629 #44544826