direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C14×He3, C32⋊2C42, 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
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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