D₃×#19128628 #44544825

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

Aliases

direct product

D₃×#19128628

Splittings

Fitting, socle, derived	C₃×C₇×C₁₂₇ ⋊ C₂²×C₃
Hall	C₃×#14581888 ⋊ 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)(48,63,79)(51,65,76)(54,68,73)(57,71,70)(60,74,67)(62,77,64)(66,82,95)(69,84,92)(72,87,89)(75,90,86)(78,93,83)(80,96,81)(85,99,109)(88,101,106)(91,104,103)(94,107,100)(97,110,98)(102,113,120)(105,115,117)(108,118,114)(111,121,112)(116,124,128)(119,126,125)(122,129,123)(127,132,133)(130,134,131)(135,137,136);(2,3)(4,5,7,10,9,8,6)(11,12,14,18,25,35,48,63,80,97,111,122,130,135,134,118,90,56,30,15,19,26,36,49,64,81,82,47,62,78,94,108,119,127,129,107,74,43,22,29,39,52,67,83,98,99,65,34,46,60,75,91,105,116,121,93,59,32,42,55,70,86,100,112,113,84,50,24,33,44,57,72,88,102,110,77,45,58,73,89,103,114,123,124,101,68,37,17,23,31,41,54,69,85,96,61,76,92,106,117,125,131,132,115,87,53,27,13,16,21,28,38,51,66,79,95,109,120,128,133,136,137,126,104,71,40,20)

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, 207);

D3×#19128628 #44544825