metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C41:Dic3, C123:1C4, D41.S3, C3:(C41:C4), (C3xD41).1C2, SmallGroup(492,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C123 — C41:Dic3 |
Generators and relations for C41:Dic3
G = < a,b,c | a41=b6=1, c2=b3, bab-1=a-1, cac-1=a32, cbc-1=b-1 >
Quotients: C1, C2, C4, S3, Dic3, C41:C4, C41:Dic3
Smallest permutation representation of C41:Dic3
►On 123 points
Generators in S123
(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)
(1 97 42)(2 96 43 41 98 82)(3 95 44 40 99 81)(4 94 45 39 100 80)(5 93 46 38 101 79)(6 92 47 37 102 78)(7 91 48 36 103 77)(8 90 49 35 104 76)(9 89 50 34 105 75)(10 88 51 33 106 74)(11 87 52 32 107 73)(12 86 53 31 108 72)(13 85 54 30 109 71)(14 84 55 29 110 70)(15 83 56 28 111 69)(16 123 57 27 112 68)(17 122 58 26 113 67)(18 121 59 25 114 66)(19 120 60 24 115 65)(20 119 61 23 116 64)(21 118 62 22 117 63)
(2 10 41 33)(3 19 40 24)(4 28 39 15)(5 37 38 6)(7 14 36 29)(8 23 35 20)(9 32 34 11)(12 18 31 25)(13 27 30 16)(17 22 26 21)(42 97)(43 106 82 88)(44 115 81 120)(45 83 80 111)(46 92 79 102)(47 101 78 93)(48 110 77 84)(49 119 76 116)(50 87 75 107)(51 96 74 98)(52 105 73 89)(53 114 72 121)(54 123 71 112)(55 91 70 103)(56 100 69 94)(57 109 68 85)(58 118 67 117)(59 86 66 108)(60 95 65 99)(61 104 64 90)(62 113 63 122)
G:=sub<Sym(123)| (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), (1,97,42)(2,96,43,41,98,82)(3,95,44,40,99,81)(4,94,45,39,100,80)(5,93,46,38,101,79)(6,92,47,37,102,78)(7,91,48,36,103,77)(8,90,49,35,104,76)(9,89,50,34,105,75)(10,88,51,33,106,74)(11,87,52,32,107,73)(12,86,53,31,108,72)(13,85,54,30,109,71)(14,84,55,29,110,70)(15,83,56,28,111,69)(16,123,57,27,112,68)(17,122,58,26,113,67)(18,121,59,25,114,66)(19,120,60,24,115,65)(20,119,61,23,116,64)(21,118,62,22,117,63), (2,10,41,33)(3,19,40,24)(4,28,39,15)(5,37,38,6)(7,14,36,29)(8,23,35,20)(9,32,34,11)(12,18,31,25)(13,27,30,16)(17,22,26,21)(42,97)(43,106,82,88)(44,115,81,120)(45,83,80,111)(46,92,79,102)(47,101,78,93)(48,110,77,84)(49,119,76,116)(50,87,75,107)(51,96,74,98)(52,105,73,89)(53,114,72,121)(54,123,71,112)(55,91,70,103)(56,100,69,94)(57,109,68,85)(58,118,67,117)(59,86,66,108)(60,95,65,99)(61,104,64,90)(62,113,63,122)>;
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), (1,97,42)(2,96,43,41,98,82)(3,95,44,40,99,81)(4,94,45,39,100,80)(5,93,46,38,101,79)(6,92,47,37,102,78)(7,91,48,36,103,77)(8,90,49,35,104,76)(9,89,50,34,105,75)(10,88,51,33,106,74)(11,87,52,32,107,73)(12,86,53,31,108,72)(13,85,54,30,109,71)(14,84,55,29,110,70)(15,83,56,28,111,69)(16,123,57,27,112,68)(17,122,58,26,113,67)(18,121,59,25,114,66)(19,120,60,24,115,65)(20,119,61,23,116,64)(21,118,62,22,117,63), (2,10,41,33)(3,19,40,24)(4,28,39,15)(5,37,38,6)(7,14,36,29)(8,23,35,20)(9,32,34,11)(12,18,31,25)(13,27,30,16)(17,22,26,21)(42,97)(43,106,82,88)(44,115,81,120)(45,83,80,111)(46,92,79,102)(47,101,78,93)(48,110,77,84)(49,119,76,116)(50,87,75,107)(51,96,74,98)(52,105,73,89)(53,114,72,121)(54,123,71,112)(55,91,70,103)(56,100,69,94)(57,109,68,85)(58,118,67,117)(59,86,66,108)(60,95,65,99)(61,104,64,90)(62,113,63,122) );
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)], [(1,97,42),(2,96,43,41,98,82),(3,95,44,40,99,81),(4,94,45,39,100,80),(5,93,46,38,101,79),(6,92,47,37,102,78),(7,91,48,36,103,77),(8,90,49,35,104,76),(9,89,50,34,105,75),(10,88,51,33,106,74),(11,87,52,32,107,73),(12,86,53,31,108,72),(13,85,54,30,109,71),(14,84,55,29,110,70),(15,83,56,28,111,69),(16,123,57,27,112,68),(17,122,58,26,113,67),(18,121,59,25,114,66),(19,120,60,24,115,65),(20,119,61,23,116,64),(21,118,62,22,117,63)], [(2,10,41,33),(3,19,40,24),(4,28,39,15),(5,37,38,6),(7,14,36,29),(8,23,35,20),(9,32,34,11),(12,18,31,25),(13,27,30,16),(17,22,26,21),(42,97),(43,106,82,88),(44,115,81,120),(45,83,80,111),(46,92,79,102),(47,101,78,93),(48,110,77,84),(49,119,76,116),(50,87,75,107),(51,96,74,98),(52,105,73,89),(53,114,72,121),(54,123,71,112),(55,91,70,103),(56,100,69,94),(57,109,68,85),(58,118,67,117),(59,86,66,108),(60,95,65,99),(61,104,64,90),(62,113,63,122)]])
36 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 41A | ··· | 41J | 123A | ··· | 123T |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 41 | ··· | 41 | 123 | ··· | 123 |
| size | 1 | 41 | 2 | 123 | 123 | 82 | 4 | ··· | 4 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | + | ||
| image | C1 | C2 | C4 | S3 | Dic3 | C41:C4 | C41:Dic3 |
| kernel | C41:Dic3 | C3xD41 | C123 | D41 | C41 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 10 | 20 |
Matrix representation of C41:Dic3 ►in GL4(F2953) generated by
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2952 | 2644 | 554 | 2644 |
| 971 | 1126 | 1943 | 1392 |
| 2293 | 2431 | 2559 | 1982 |
| 2614 | 138 | 2165 | 660 |
| 1534 | 398 | 2056 | 339 |
| 1 | 0 | 0 | 0 |
| 848 | 229 | 234 | 1268 |
| 1038 | 780 | 2532 | 614 |
| 1939 | 1797 | 2739 | 191 |
G:=sub<GL(4,GF(2953))| [0,0,0,2952,1,0,0,2644,0,1,0,554,0,0,1,2644],[971,2293,2614,1534,1126,2431,138,398,1943,2559,2165,2056,1392,1982,660,339],[1,848,1038,1939,0,229,780,1797,0,234,2532,2739,0,1268,614,191] >;
C41:Dic3 in GAP, Magma, Sage, TeX
C_{41}\rtimes {\rm Dic}_3 % in TeX
G:=Group("C41:Dic3"); // GroupNames label
G:=SmallGroup(492,6);
// by ID
G=gap.SmallGroup(492,6);
# by ID
G:=PCGroup([4,-2,-2,-3,-41,8,98,1731,3847]);
// Polycyclic
G:=Group<a,b,c|a^41=b^6=1,c^2=b^3,b*a*b^-1=a^-1,c*a*c^-1=a^32,c*b*c^-1=b^-1>;
// generators/relations
Export