direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5xD23, C23:C10, C115:2C2, SmallGroup(230,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — C5xD23 |
Generators and relations for C5xD23
G = < a,b,c | a5=b23=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C5, C10, D23, C5xD23
Smallest permutation representation of C5xD23
►On 115 points
Generators in S115
(1 107 84 50 40)(2 108 85 51 41)(3 109 86 52 42)(4 110 87 53 43)(5 111 88 54 44)(6 112 89 55 45)(7 113 90 56 46)(8 114 91 57 24)(9 115 92 58 25)(10 93 70 59 26)(11 94 71 60 27)(12 95 72 61 28)(13 96 73 62 29)(14 97 74 63 30)(15 98 75 64 31)(16 99 76 65 32)(17 100 77 66 33)(18 101 78 67 34)(19 102 79 68 35)(20 103 80 69 36)(21 104 81 47 37)(22 105 82 48 38)(23 106 83 49 39)
(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)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(24 32)(25 31)(26 30)(27 29)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(47 52)(48 51)(49 50)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(70 74)(71 73)(75 92)(76 91)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)(93 97)(94 96)(98 115)(99 114)(100 113)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)
G:=sub<Sym(115)| (1,107,84,50,40)(2,108,85,51,41)(3,109,86,52,42)(4,110,87,53,43)(5,111,88,54,44)(6,112,89,55,45)(7,113,90,56,46)(8,114,91,57,24)(9,115,92,58,25)(10,93,70,59,26)(11,94,71,60,27)(12,95,72,61,28)(13,96,73,62,29)(14,97,74,63,30)(15,98,75,64,31)(16,99,76,65,32)(17,100,77,66,33)(18,101,78,67,34)(19,102,79,68,35)(20,103,80,69,36)(21,104,81,47,37)(22,105,82,48,38)(23,106,83,49,39), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,32)(25,31)(26,30)(27,29)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(47,52)(48,51)(49,50)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(70,74)(71,73)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(93,97)(94,96)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)>;
G:=Group( (1,107,84,50,40)(2,108,85,51,41)(3,109,86,52,42)(4,110,87,53,43)(5,111,88,54,44)(6,112,89,55,45)(7,113,90,56,46)(8,114,91,57,24)(9,115,92,58,25)(10,93,70,59,26)(11,94,71,60,27)(12,95,72,61,28)(13,96,73,62,29)(14,97,74,63,30)(15,98,75,64,31)(16,99,76,65,32)(17,100,77,66,33)(18,101,78,67,34)(19,102,79,68,35)(20,103,80,69,36)(21,104,81,47,37)(22,105,82,48,38)(23,106,83,49,39), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,32)(25,31)(26,30)(27,29)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(47,52)(48,51)(49,50)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(70,74)(71,73)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(93,97)(94,96)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107) );
G=PermutationGroup([[(1,107,84,50,40),(2,108,85,51,41),(3,109,86,52,42),(4,110,87,53,43),(5,111,88,54,44),(6,112,89,55,45),(7,113,90,56,46),(8,114,91,57,24),(9,115,92,58,25),(10,93,70,59,26),(11,94,71,60,27),(12,95,72,61,28),(13,96,73,62,29),(14,97,74,63,30),(15,98,75,64,31),(16,99,76,65,32),(17,100,77,66,33),(18,101,78,67,34),(19,102,79,68,35),(20,103,80,69,36),(21,104,81,47,37),(22,105,82,48,38),(23,106,83,49,39)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13),(24,32),(25,31),(26,30),(27,29),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(47,52),(48,51),(49,50),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62),(70,74),(71,73),(75,92),(76,91),(77,90),(78,89),(79,88),(80,87),(81,86),(82,85),(83,84),(93,97),(94,96),(98,115),(99,114),(100,113),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107)]])
65 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 23A | ··· | 23K | 115A | ··· | 115AR |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 23 | ··· | 23 | 115 | ··· | 115 |
| size | 1 | 23 | 1 | 1 | 1 | 1 | 23 | 23 | 23 | 23 | 2 | ··· | 2 | 2 | ··· | 2 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C5 | C10 | D23 | C5xD23 |
| kernel | C5xD23 | C115 | D23 | C23 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 11 | 44 |
Matrix representation of C5xD23 ►in GL2(F461) generated by
| 368 | 0 |
| 0 | 368 |
| 0 | 1 |
| 460 | 418 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(461))| [368,0,0,368],[0,460,1,418],[0,1,1,0] >;
C5xD23 in GAP, Magma, Sage, TeX
C_5\times D_{23} % in TeX
G:=Group("C5xD23"); // GroupNames label
G:=SmallGroup(230,2);
// by ID
G=gap.SmallGroup(230,2);
# by ID
G:=PCGroup([3,-2,-5,-23,1982]);
// Polycyclic
G:=Group<a,b,c|a^5=b^23=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export