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## G = C14×3- 1+2order 378 = 2·33·7

### Direct product of C14 and 3- 1+2

direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary

Aliases: C14×3- 1+2, C18⋊C21, C92C42, C6316C6, C1264C3, C32.C42, C42.8C32, (C3×C6).C21, C3.2(C3×C42), (C3×C42).1C3, C6.2(C3×C21), (C3×C21).6C6, C21.17(C3×C6), SmallGroup(378,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C14×3- 1+2
 Chief series C1 — C3 — C21 — C3×C21 — C7×3- 1+2 — C14×3- 1+2
 Lower central C1 — C3 — C14×3- 1+2
 Upper central C1 — C42 — C14×3- 1+2

Generators and relations for C14×3- 1+2
G = < a,b,c | a14=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C14×3- 1+2
On 126 points
Generators in S126
(1 2 3 4 5 6 7 8 9 10 11 12 13 14)(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)
(1 117 33 98 108 79 54 61 22)(2 118 34 85 109 80 55 62 23)(3 119 35 86 110 81 56 63 24)(4 120 36 87 111 82 43 64 25)(5 121 37 88 112 83 44 65 26)(6 122 38 89 99 84 45 66 27)(7 123 39 90 100 71 46 67 28)(8 124 40 91 101 72 47 68 15)(9 125 41 92 102 73 48 69 16)(10 126 42 93 103 74 49 70 17)(11 113 29 94 104 75 50 57 18)(12 114 30 95 105 76 51 58 19)(13 115 31 96 106 77 52 59 20)(14 116 32 97 107 78 53 60 21)
(15 40 72)(16 41 73)(17 42 74)(18 29 75)(19 30 76)(20 31 77)(21 32 78)(22 33 79)(23 34 80)(24 35 81)(25 36 82)(26 37 83)(27 38 84)(28 39 71)(57 104 113)(58 105 114)(59 106 115)(60 107 116)(61 108 117)(62 109 118)(63 110 119)(64 111 120)(65 112 121)(66 99 122)(67 100 123)(68 101 124)(69 102 125)(70 103 126)

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

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

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

154 conjugacy classes

 class 1 2 3A 3B 3C 3D 6A 6B 6C 6D 7A ··· 7F 9A ··· 9F 14A ··· 14F 18A ··· 18F 21A ··· 21L 21M ··· 21X 42A ··· 42L 42M ··· 42X 63A ··· 63AJ 126A ··· 126AJ order 1 2 3 3 3 3 6 6 6 6 7 ··· 7 9 ··· 9 14 ··· 14 18 ··· 18 21 ··· 21 21 ··· 21 42 ··· 42 42 ··· 42 63 ··· 63 126 ··· 126 size 1 1 1 1 3 3 1 1 3 3 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C6 C6 C7 C14 C21 C21 C42 C42 3- 1+2 C2×3- 1+2 C7×3- 1+2 C14×3- 1+2 kernel C14×3- 1+2 C7×3- 1+2 C126 C3×C42 C63 C3×C21 C2×3- 1+2 3- 1+2 C18 C3×C6 C9 C32 C14 C7 C2 C1 # reps 1 1 6 2 6 2 6 6 36 12 36 12 2 2 12 12

Matrix representation of C14×3- 1+2 in GL3(𝔽127) generated by

 125 0 0 0 125 0 0 0 125
,
 0 1 0 0 0 19 1 0 0
,
 1 0 0 0 19 0 0 0 107
G:=sub<GL(3,GF(127))| [125,0,0,0,125,0,0,0,125],[0,0,1,1,0,0,0,19,0],[1,0,0,0,19,0,0,0,107] >;

C14×3- 1+2 in GAP, Magma, Sage, TeX

C_{14}\times 3_-^{1+2}
% in TeX

G:=Group("C14xES-(3,1)");
// GroupNames label

G:=SmallGroup(378,46);
// by ID

G=gap.SmallGroup(378,46);
# by ID

G:=PCGroup([5,-2,-3,-3,-7,-3,636,997]);
// Polycyclic

G:=Group<a,b,c|a^14=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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