direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D5×C23, C5⋊C46, C115⋊3C2, SmallGroup(230,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C23 |
Generators and relations for D5×C23
G = < a,b,c | a23=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C23
►On 115 points
Generators in S115
(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 36 78 94 64)(2 37 79 95 65)(3 38 80 96 66)(4 39 81 97 67)(5 40 82 98 68)(6 41 83 99 69)(7 42 84 100 47)(8 43 85 101 48)(9 44 86 102 49)(10 45 87 103 50)(11 46 88 104 51)(12 24 89 105 52)(13 25 90 106 53)(14 26 91 107 54)(15 27 92 108 55)(16 28 70 109 56)(17 29 71 110 57)(18 30 72 111 58)(19 31 73 112 59)(20 32 74 113 60)(21 33 75 114 61)(22 34 76 115 62)(23 35 77 93 63)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 105)(25 106)(26 107)(27 108)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)
G:=sub<Sym(115)| (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), (1,36,78,94,64)(2,37,79,95,65)(3,38,80,96,66)(4,39,81,97,67)(5,40,82,98,68)(6,41,83,99,69)(7,42,84,100,47)(8,43,85,101,48)(9,44,86,102,49)(10,45,87,103,50)(11,46,88,104,51)(12,24,89,105,52)(13,25,90,106,53)(14,26,91,107,54)(15,27,92,108,55)(16,28,70,109,56)(17,29,71,110,57)(18,30,72,111,58)(19,31,73,112,59)(20,32,74,113,60)(21,33,75,114,61)(22,34,76,115,62)(23,35,77,93,63), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)>;
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), (1,36,78,94,64)(2,37,79,95,65)(3,38,80,96,66)(4,39,81,97,67)(5,40,82,98,68)(6,41,83,99,69)(7,42,84,100,47)(8,43,85,101,48)(9,44,86,102,49)(10,45,87,103,50)(11,46,88,104,51)(12,24,89,105,52)(13,25,90,106,53)(14,26,91,107,54)(15,27,92,108,55)(16,28,70,109,56)(17,29,71,110,57)(18,30,72,111,58)(19,31,73,112,59)(20,32,74,113,60)(21,33,75,114,61)(22,34,76,115,62)(23,35,77,93,63), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104) );
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)], [(1,36,78,94,64),(2,37,79,95,65),(3,38,80,96,66),(4,39,81,97,67),(5,40,82,98,68),(6,41,83,99,69),(7,42,84,100,47),(8,43,85,101,48),(9,44,86,102,49),(10,45,87,103,50),(11,46,88,104,51),(12,24,89,105,52),(13,25,90,106,53),(14,26,91,107,54),(15,27,92,108,55),(16,28,70,109,56),(17,29,71,110,57),(18,30,72,111,58),(19,31,73,112,59),(20,32,74,113,60),(21,33,75,114,61),(22,34,76,115,62),(23,35,77,93,63)], [(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,105),(25,106),(26,107),(27,108),(28,109),(29,110),(30,111),(31,112),(32,113),(33,114),(34,115),(35,93),(36,94),(37,95),(38,96),(39,97),(40,98),(41,99),(42,100),(43,101),(44,102),(45,103),(46,104)]])
D5×C23 is a maximal subgroup of
C23⋊F5
92 conjugacy classes
| class | 1 | 2 | 5A | 5B | 23A | ··· | 23V | 46A | ··· | 46V | 115A | ··· | 115AR |
| order | 1 | 2 | 5 | 5 | 23 | ··· | 23 | 46 | ··· | 46 | 115 | ··· | 115 |
| size | 1 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C23 | C46 | D5 | D5×C23 |
| kernel | D5×C23 | C115 | D5 | C5 | C23 | C1 |
| # reps | 1 | 1 | 22 | 22 | 2 | 44 |
Matrix representation of D5×C23 ►in GL2(𝔽461) generated by
| 441 | 0 |
| 0 | 441 |
| 21 | 1 |
| 460 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(461))| [441,0,0,441],[21,460,1,0],[0,1,1,0] >;
D5×C23 in GAP, Magma, Sage, TeX
D_5\times C_{23} % in TeX
G:=Group("D5xC23"); // GroupNames label
G:=SmallGroup(230,1);
// by ID
G=gap.SmallGroup(230,1);
# by ID
G:=PCGroup([3,-2,-23,-5,1658]);
// Polycyclic
G:=Group<a,b,c|a^23=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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