D₃×#19128624 #44544830

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

Aliases

direct product

D₃×#19128624

Splittings

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

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, 212);

D3×#19128624 #44544830