C₇×C₃⋊D₃×D₁₃₇ #46828666

order	34524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₇×C₃⋊D₃×D₁₃₇
unambiguous semidirect product	(C₃×D₁₃₇)⋊(D₃×C₇)

Splittings

Fitting, socle, Hall	C₃²×C₇×C₁₃₇ ⋊ C₂²
derived, Hall	C₃²×C₁₃₇ ⋊ C₂²×C₇
Hall	C₇×C₁₃₇ ⋊ C₂×C₃⋊D₃
Sylow	C₁₃₇ ⋊ C₂×C₇×C₃⋊D₃
Hall	C₃²×C₇ ⋊ C₂×D₁₃₇
Sylow	C₃² ⋊ C₂×C₇×D₁₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	C₃⋊D₃×D₁₃₇
commutator, cosocle, hypocenter	C₃²×C₁₃₇	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₁₃₇	C₂²
LCS	G ▷ C₃²×C₁₃₇
UCS	C₁ ◁ C₇

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 150
transitive degree	μ_t(G) = 8631

A minimal faithful permutation representation — 3 generators on 150 points:

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

Conjugacy classes & character table

2940 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34524, 109);

C7×C3⋊D3×D137 #46828666