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## G = C14×He3order 378 = 2·33·7

### Direct product of C14 and He3

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

Aliases: C14×He3, C322C42, C42.7C32, (C3×C6)⋊C21, (C3×C42)⋊1C3, (C3×C21)⋊12C6, C3.1(C3×C42), C6.1(C3×C21), C21.16(C3×C6), SmallGroup(378,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C14×He3
 Chief series C1 — C3 — C21 — C3×C21 — C7×He3 — C14×He3
 Lower central C1 — C3 — C14×He3
 Upper central C1 — C42 — C14×He3

Generators and relations for C14×He3
G = < a,b,c,d | a14=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Smallest permutation representation of C14×He3
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)
(15 60 117)(16 61 118)(17 62 119)(18 63 120)(19 64 121)(20 65 122)(21 66 123)(22 67 124)(23 68 125)(24 69 126)(25 70 113)(26 57 114)(27 58 115)(28 59 116)(29 86 79)(30 87 80)(31 88 81)(32 89 82)(33 90 83)(34 91 84)(35 92 71)(36 93 72)(37 94 73)(38 95 74)(39 96 75)(40 97 76)(41 98 77)(42 85 78)
(1 105 52)(2 106 53)(3 107 54)(4 108 55)(5 109 56)(6 110 43)(7 111 44)(8 112 45)(9 99 46)(10 100 47)(11 101 48)(12 102 49)(13 103 50)(14 104 51)(15 60 117)(16 61 118)(17 62 119)(18 63 120)(19 64 121)(20 65 122)(21 66 123)(22 67 124)(23 68 125)(24 69 126)(25 70 113)(26 57 114)(27 58 115)(28 59 116)(29 79 86)(30 80 87)(31 81 88)(32 82 89)(33 83 90)(34 84 91)(35 71 92)(36 72 93)(37 73 94)(38 74 95)(39 75 96)(40 76 97)(41 77 98)(42 78 85)
(1 96 116)(2 97 117)(3 98 118)(4 85 119)(5 86 120)(6 87 121)(7 88 122)(8 89 123)(9 90 124)(10 91 125)(11 92 126)(12 93 113)(13 94 114)(14 95 115)(15 106 40)(16 107 41)(17 108 42)(18 109 29)(19 110 30)(20 111 31)(21 112 32)(22 99 33)(23 100 34)(24 101 35)(25 102 36)(26 103 37)(27 104 38)(28 105 39)(43 80 64)(44 81 65)(45 82 66)(46 83 67)(47 84 68)(48 71 69)(49 72 70)(50 73 57)(51 74 58)(52 75 59)(53 76 60)(54 77 61)(55 78 62)(56 79 63)

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

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

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

154 conjugacy classes

 class 1 2 3A 3B 3C ··· 3J 6A 6B 6C ··· 6J 7A ··· 7F 14A ··· 14F 21A ··· 21L 21M ··· 21BH 42A ··· 42L 42M ··· 42BH order 1 2 3 3 3 ··· 3 6 6 6 ··· 6 7 ··· 7 14 ··· 14 21 ··· 21 21 ··· 21 42 ··· 42 42 ··· 42 size 1 1 1 1 3 ··· 3 1 1 3 ··· 3 1 ··· 1 1 ··· 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C6 C7 C14 C21 C42 He3 C2×He3 C7×He3 C14×He3 kernel C14×He3 C7×He3 C3×C42 C3×C21 C2×He3 He3 C3×C6 C32 C14 C7 C2 C1 # reps 1 1 8 8 6 6 48 48 2 2 12 12

Matrix representation of C14×He3 in GL4(𝔽43) generated by

 42 0 0 0 0 11 0 0 0 0 11 0 0 0 0 11
,
 6 0 0 0 0 1 0 0 0 0 36 0 0 0 0 6
,
 1 0 0 0 0 36 0 0 0 0 36 0 0 0 0 36
,
 36 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
G:=sub<GL(4,GF(43))| [42,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[6,0,0,0,0,1,0,0,0,0,36,0,0,0,0,6],[1,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

C14×He3 in GAP, Magma, Sage, TeX

C_{14}\times {\rm He}_3
% in TeX

G:=Group("C14xHe3");
// GroupNames label

G:=SmallGroup(378,45);
// by ID

G=gap.SmallGroup(378,45);
# by ID

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

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

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