D₃×#19128625 #44544831

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

Aliases

direct product

D₃×#19128625

Splittings

Fitting, socle, derived	C₃×C₇×C₁₂₇ ⋊ C₂²×C₃
Hall	C₃×#14581887 ⋊ C₂²
Hall	C₇×C₁₂₇ ⋊ C₂×C₃×D₃
Sylow	C₁₂₇ ⋊ C₂×D₃×ΓL(1,𝔽₈)
Sylow	C₇ ⋊ D₃×D₁₂₇⋊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)(5,6,7)(8,10,9)(12,16,17,13,14,18)(15,24,25,19,20,34)(21,37,38,27,28,51)(22,36,32,30,26,23)(29,56,57,40,41,74)(31,54,55,43,39,48)(33,35,53,45,46,49)(42,79,80,59,60,103)(44,77,78,62,58,71)(47,52,76,64,65,72)(50,92,90,68,69,67)(61,108,84,82,83,110)(63,106,107,85,81,100)(66,75,105,87,88,101)(70,118,119,93,94,96)(73,123,116,97,98,91)(86,131,115,112,109,129)(89,104,111,113,114,130)(95,135,134,120,121,127)(99,137,136,124,125,122)(102,132,133,126,128,117);(2,3)(4,5,6,8,7,9,10)(11,12,15,21,29,42,61,84,111,132,137,135,131,108,112,85,62,43,30,14,18,23,33,47,66,89,115,80,105,123,118,106,79,104,120,93,68,46,20,26,34,48,67,91,117,134,107,57,76,92,77,56,75,102,124,97,65,28,39,49,51,71,96,122,136,119,78,38,53,54,37,52,73,99,126,88,41,58,69,72,74,100,127,133,116,90,55,25,36,24,35,50,70,95,114,60,81,94,98,101,103,129,113,87,64,45,32,17,16,22,31,44,63,86,83,109,121,125,128,130,110,82,59,40,27,19,13)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32004, 213);

D3×#19128625 #44544831