direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×C37⋊C4, C111⋊2C4, C37⋊3C12, D37.2C6, (C3×D37).2C2, SmallGroup(444,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C3×C37⋊C4 |
Generators and relations for C3×C37⋊C4
G = < a,b,c | a3=b37=c4=1, ab=ba, ac=ca, cbc-1=b6 >
Smallest permutation representation of C3×C37⋊C4
►On 111 points
Generators in S111
(1 75 38)(2 76 39)(3 77 40)(4 78 41)(5 79 42)(6 80 43)(7 81 44)(8 82 45)(9 83 46)(10 84 47)(11 85 48)(12 86 49)(13 87 50)(14 88 51)(15 89 52)(16 90 53)(17 91 54)(18 92 55)(19 93 56)(20 94 57)(21 95 58)(22 96 59)(23 97 60)(24 98 61)(25 99 62)(26 100 63)(27 101 64)(28 102 65)(29 103 66)(30 104 67)(31 105 68)(32 106 69)(33 107 70)(34 108 71)(35 109 72)(36 110 73)(37 111 74)
(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)
(2 32 37 7)(3 26 36 13)(4 20 35 19)(5 14 34 25)(6 8 33 31)(9 27 30 12)(10 21 29 18)(11 15 28 24)(16 22 23 17)(39 69 74 44)(40 63 73 50)(41 57 72 56)(42 51 71 62)(43 45 70 68)(46 64 67 49)(47 58 66 55)(48 52 65 61)(53 59 60 54)(76 106 111 81)(77 100 110 87)(78 94 109 93)(79 88 108 99)(80 82 107 105)(83 101 104 86)(84 95 103 92)(85 89 102 98)(90 96 97 91)
G:=sub<Sym(111)| (1,75,38)(2,76,39)(3,77,40)(4,78,41)(5,79,42)(6,80,43)(7,81,44)(8,82,45)(9,83,46)(10,84,47)(11,85,48)(12,86,49)(13,87,50)(14,88,51)(15,89,52)(16,90,53)(17,91,54)(18,92,55)(19,93,56)(20,94,57)(21,95,58)(22,96,59)(23,97,60)(24,98,61)(25,99,62)(26,100,63)(27,101,64)(28,102,65)(29,103,66)(30,104,67)(31,105,68)(32,106,69)(33,107,70)(34,108,71)(35,109,72)(36,110,73)(37,111,74), (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), (2,32,37,7)(3,26,36,13)(4,20,35,19)(5,14,34,25)(6,8,33,31)(9,27,30,12)(10,21,29,18)(11,15,28,24)(16,22,23,17)(39,69,74,44)(40,63,73,50)(41,57,72,56)(42,51,71,62)(43,45,70,68)(46,64,67,49)(47,58,66,55)(48,52,65,61)(53,59,60,54)(76,106,111,81)(77,100,110,87)(78,94,109,93)(79,88,108,99)(80,82,107,105)(83,101,104,86)(84,95,103,92)(85,89,102,98)(90,96,97,91)>;
G:=Group( (1,75,38)(2,76,39)(3,77,40)(4,78,41)(5,79,42)(6,80,43)(7,81,44)(8,82,45)(9,83,46)(10,84,47)(11,85,48)(12,86,49)(13,87,50)(14,88,51)(15,89,52)(16,90,53)(17,91,54)(18,92,55)(19,93,56)(20,94,57)(21,95,58)(22,96,59)(23,97,60)(24,98,61)(25,99,62)(26,100,63)(27,101,64)(28,102,65)(29,103,66)(30,104,67)(31,105,68)(32,106,69)(33,107,70)(34,108,71)(35,109,72)(36,110,73)(37,111,74), (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), (2,32,37,7)(3,26,36,13)(4,20,35,19)(5,14,34,25)(6,8,33,31)(9,27,30,12)(10,21,29,18)(11,15,28,24)(16,22,23,17)(39,69,74,44)(40,63,73,50)(41,57,72,56)(42,51,71,62)(43,45,70,68)(46,64,67,49)(47,58,66,55)(48,52,65,61)(53,59,60,54)(76,106,111,81)(77,100,110,87)(78,94,109,93)(79,88,108,99)(80,82,107,105)(83,101,104,86)(84,95,103,92)(85,89,102,98)(90,96,97,91) );
G=PermutationGroup([[(1,75,38),(2,76,39),(3,77,40),(4,78,41),(5,79,42),(6,80,43),(7,81,44),(8,82,45),(9,83,46),(10,84,47),(11,85,48),(12,86,49),(13,87,50),(14,88,51),(15,89,52),(16,90,53),(17,91,54),(18,92,55),(19,93,56),(20,94,57),(21,95,58),(22,96,59),(23,97,60),(24,98,61),(25,99,62),(26,100,63),(27,101,64),(28,102,65),(29,103,66),(30,104,67),(31,105,68),(32,106,69),(33,107,70),(34,108,71),(35,109,72),(36,110,73),(37,111,74)], [(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)], [(2,32,37,7),(3,26,36,13),(4,20,35,19),(5,14,34,25),(6,8,33,31),(9,27,30,12),(10,21,29,18),(11,15,28,24),(16,22,23,17),(39,69,74,44),(40,63,73,50),(41,57,72,56),(42,51,71,62),(43,45,70,68),(46,64,67,49),(47,58,66,55),(48,52,65,61),(53,59,60,54),(76,106,111,81),(77,100,110,87),(78,94,109,93),(79,88,108,99),(80,82,107,105),(83,101,104,86),(84,95,103,92),(85,89,102,98),(90,96,97,91)]])
39 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 37A | ··· | 37I | 111A | ··· | 111R |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 37 | ··· | 37 | 111 | ··· | 111 |
| size | 1 | 37 | 1 | 1 | 37 | 37 | 37 | 37 | 37 | 37 | 37 | 37 | 4 | ··· | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | |||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | C37⋊C4 | C3×C37⋊C4 |
| kernel | C3×C37⋊C4 | C3×D37 | C37⋊C4 | C111 | D37 | C37 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 9 | 18 |
Matrix representation of C3×C37⋊C4 ►in GL5(𝔽1777)
| 1147 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 761 | 669 | 761 | 1776 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 775 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 1289 | 991 | 1099 | 1623 |
| 0 | 631 | 1285 | 247 | 190 |
| 0 | 1170 | 577 | 1018 | 538 |
G:=sub<GL(5,GF(1777))| [1147,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,761,1,0,0,0,669,0,1,0,0,761,0,0,1,0,1776,0,0,0],[775,0,0,0,0,0,1,1289,631,1170,0,0,991,1285,577,0,0,1099,247,1018,0,0,1623,190,538] >;
C3×C37⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_{37}\rtimes C_4 % in TeX
G:=Group("C3xC37:C4"); // GroupNames label
G:=SmallGroup(444,9);
// by ID
G=gap.SmallGroup(444,9);
# by ID
G:=PCGroup([4,-2,-3,-2,-37,24,5955,1163]);
// Polycyclic
G:=Group<a,b,c|a^3=b^37=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
// generators/relations
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