direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×C41⋊C4, C41⋊C12, C123⋊2C4, D41.C6, (C3×D41).2C2, SmallGroup(492,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C41 — C3×C41⋊C4 |
Generators and relations for C3×C41⋊C4
G = < a,b,c | a3=b41=c4=1, ab=ba, ac=ca, cbc-1=b9 >
Smallest permutation representation of C3×C41⋊C4
►On 123 points
Generators in S123
(1 83 42)(2 84 43)(3 85 44)(4 86 45)(5 87 46)(6 88 47)(7 89 48)(8 90 49)(9 91 50)(10 92 51)(11 93 52)(12 94 53)(13 95 54)(14 96 55)(15 97 56)(16 98 57)(17 99 58)(18 100 59)(19 101 60)(20 102 61)(21 103 62)(22 104 63)(23 105 64)(24 106 65)(25 107 66)(26 108 67)(27 109 68)(28 110 69)(29 111 70)(30 112 71)(31 113 72)(32 114 73)(33 115 74)(34 116 75)(35 117 76)(36 118 77)(37 119 78)(38 120 79)(39 121 80)(40 122 81)(41 123 82)
(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)
(2 33 41 10)(3 24 40 19)(4 15 39 28)(5 6 38 37)(7 29 36 14)(8 20 35 23)(9 11 34 32)(12 25 31 18)(13 16 30 27)(17 21 26 22)(43 74 82 51)(44 65 81 60)(45 56 80 69)(46 47 79 78)(48 70 77 55)(49 61 76 64)(50 52 75 73)(53 66 72 59)(54 57 71 68)(58 62 67 63)(84 115 123 92)(85 106 122 101)(86 97 121 110)(87 88 120 119)(89 111 118 96)(90 102 117 105)(91 93 116 114)(94 107 113 100)(95 98 112 109)(99 103 108 104)
G:=sub<Sym(123)| (1,83,42)(2,84,43)(3,85,44)(4,86,45)(5,87,46)(6,88,47)(7,89,48)(8,90,49)(9,91,50)(10,92,51)(11,93,52)(12,94,53)(13,95,54)(14,96,55)(15,97,56)(16,98,57)(17,99,58)(18,100,59)(19,101,60)(20,102,61)(21,103,62)(22,104,63)(23,105,64)(24,106,65)(25,107,66)(26,108,67)(27,109,68)(28,110,69)(29,111,70)(30,112,71)(31,113,72)(32,114,73)(33,115,74)(34,116,75)(35,117,76)(36,118,77)(37,119,78)(38,120,79)(39,121,80)(40,122,81)(41,123,82), (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), (2,33,41,10)(3,24,40,19)(4,15,39,28)(5,6,38,37)(7,29,36,14)(8,20,35,23)(9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22)(43,74,82,51)(44,65,81,60)(45,56,80,69)(46,47,79,78)(48,70,77,55)(49,61,76,64)(50,52,75,73)(53,66,72,59)(54,57,71,68)(58,62,67,63)(84,115,123,92)(85,106,122,101)(86,97,121,110)(87,88,120,119)(89,111,118,96)(90,102,117,105)(91,93,116,114)(94,107,113,100)(95,98,112,109)(99,103,108,104)>;
G:=Group( (1,83,42)(2,84,43)(3,85,44)(4,86,45)(5,87,46)(6,88,47)(7,89,48)(8,90,49)(9,91,50)(10,92,51)(11,93,52)(12,94,53)(13,95,54)(14,96,55)(15,97,56)(16,98,57)(17,99,58)(18,100,59)(19,101,60)(20,102,61)(21,103,62)(22,104,63)(23,105,64)(24,106,65)(25,107,66)(26,108,67)(27,109,68)(28,110,69)(29,111,70)(30,112,71)(31,113,72)(32,114,73)(33,115,74)(34,116,75)(35,117,76)(36,118,77)(37,119,78)(38,120,79)(39,121,80)(40,122,81)(41,123,82), (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), (2,33,41,10)(3,24,40,19)(4,15,39,28)(5,6,38,37)(7,29,36,14)(8,20,35,23)(9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22)(43,74,82,51)(44,65,81,60)(45,56,80,69)(46,47,79,78)(48,70,77,55)(49,61,76,64)(50,52,75,73)(53,66,72,59)(54,57,71,68)(58,62,67,63)(84,115,123,92)(85,106,122,101)(86,97,121,110)(87,88,120,119)(89,111,118,96)(90,102,117,105)(91,93,116,114)(94,107,113,100)(95,98,112,109)(99,103,108,104) );
G=PermutationGroup([[(1,83,42),(2,84,43),(3,85,44),(4,86,45),(5,87,46),(6,88,47),(7,89,48),(8,90,49),(9,91,50),(10,92,51),(11,93,52),(12,94,53),(13,95,54),(14,96,55),(15,97,56),(16,98,57),(17,99,58),(18,100,59),(19,101,60),(20,102,61),(21,103,62),(22,104,63),(23,105,64),(24,106,65),(25,107,66),(26,108,67),(27,109,68),(28,110,69),(29,111,70),(30,112,71),(31,113,72),(32,114,73),(33,115,74),(34,116,75),(35,117,76),(36,118,77),(37,119,78),(38,120,79),(39,121,80),(40,122,81),(41,123,82)], [(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)], [(2,33,41,10),(3,24,40,19),(4,15,39,28),(5,6,38,37),(7,29,36,14),(8,20,35,23),(9,11,34,32),(12,25,31,18),(13,16,30,27),(17,21,26,22),(43,74,82,51),(44,65,81,60),(45,56,80,69),(46,47,79,78),(48,70,77,55),(49,61,76,64),(50,52,75,73),(53,66,72,59),(54,57,71,68),(58,62,67,63),(84,115,123,92),(85,106,122,101),(86,97,121,110),(87,88,120,119),(89,111,118,96),(90,102,117,105),(91,93,116,114),(94,107,113,100),(95,98,112,109),(99,103,108,104)]])
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 41A | ··· | 41J | 123A | ··· | 123T |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 41 | ··· | 41 | 123 | ··· | 123 |
| size | 1 | 41 | 1 | 1 | 41 | 41 | 41 | 41 | 41 | 41 | 41 | 41 | 4 | ··· | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | |||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | C41⋊C4 | C3×C41⋊C4 |
| kernel | C3×C41⋊C4 | C3×D41 | C41⋊C4 | C123 | D41 | C41 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 10 | 20 |
Matrix representation of C3×C41⋊C4 ►in GL4(𝔽2953) generated by
| 800 | 0 | 0 | 0 |
| 0 | 800 | 0 | 0 |
| 0 | 0 | 800 | 0 |
| 0 | 0 | 0 | 800 |
| 0 | 0 | 0 | 2952 |
| 1 | 0 | 0 | 2095 |
| 0 | 1 | 0 | 2046 |
| 0 | 0 | 1 | 2095 |
| 1 | 1718 | 2312 | 2538 |
| 0 | 912 | 2874 | 2478 |
| 0 | 494 | 2298 | 1354 |
| 0 | 272 | 1224 | 2695 |
G:=sub<GL(4,GF(2953))| [800,0,0,0,0,800,0,0,0,0,800,0,0,0,0,800],[0,1,0,0,0,0,1,0,0,0,0,1,2952,2095,2046,2095],[1,0,0,0,1718,912,494,272,2312,2874,2298,1224,2538,2478,1354,2695] >;
C3×C41⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_{41}\rtimes C_4 % in TeX
G:=Group("C3xC41:C4"); // GroupNames label
G:=SmallGroup(492,5);
// by ID
G=gap.SmallGroup(492,5);
# by ID
G:=PCGroup([4,-2,-3,-2,-41,24,6147,1291]);
// Polycyclic
G:=Group<a,b,c|a^3=b^41=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations
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