direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×D23, C5⋊1D46, D115⋊C2, C23⋊1D10, C115⋊C22, (C5×D23)⋊C2, (D5×C23)⋊C2, SmallGroup(460,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C115 — D5×D23 |
Generators and relations for D5×D23
G = < a,b,c,d | a5=b2=c23=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of D5×D23
►On 115 points
Generators in S115
(1 109 82 52 40)(2 110 83 53 41)(3 111 84 54 42)(4 112 85 55 43)(5 113 86 56 44)(6 114 87 57 45)(7 115 88 58 46)(8 93 89 59 24)(9 94 90 60 25)(10 95 91 61 26)(11 96 92 62 27)(12 97 70 63 28)(13 98 71 64 29)(14 99 72 65 30)(15 100 73 66 31)(16 101 74 67 32)(17 102 75 68 33)(18 103 76 69 34)(19 104 77 47 35)(20 105 78 48 36)(21 106 79 49 37)(22 107 80 50 38)(23 108 81 51 39)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 93)(60 94)(61 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 101)(68 102)(69 103)
(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 56)(48 55)(49 54)(50 53)(51 52)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 84)(80 83)(81 82)(93 101)(94 100)(95 99)(96 98)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)
G:=sub<Sym(115)| (1,109,82,52,40)(2,110,83,53,41)(3,111,84,54,42)(4,112,85,55,43)(5,113,86,56,44)(6,114,87,57,45)(7,115,88,58,46)(8,93,89,59,24)(9,94,90,60,25)(10,95,91,61,26)(11,96,92,62,27)(12,97,70,63,28)(13,98,71,64,29)(14,99,72,65,30)(15,100,73,66,31)(16,101,74,67,32)(17,102,75,68,33)(18,103,76,69,34)(19,104,77,47,35)(20,105,78,48,36)(21,106,79,49,37)(22,107,80,50,38)(23,108,81,51,39), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,93)(60,94)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,101)(68,102)(69,103), (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)(47,56)(48,55)(49,54)(50,53)(51,52)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)(93,101)(94,100)(95,99)(96,98)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)>;
G:=Group( (1,109,82,52,40)(2,110,83,53,41)(3,111,84,54,42)(4,112,85,55,43)(5,113,86,56,44)(6,114,87,57,45)(7,115,88,58,46)(8,93,89,59,24)(9,94,90,60,25)(10,95,91,61,26)(11,96,92,62,27)(12,97,70,63,28)(13,98,71,64,29)(14,99,72,65,30)(15,100,73,66,31)(16,101,74,67,32)(17,102,75,68,33)(18,103,76,69,34)(19,104,77,47,35)(20,105,78,48,36)(21,106,79,49,37)(22,107,80,50,38)(23,108,81,51,39), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,93)(60,94)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,101)(68,102)(69,103), (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)(47,56)(48,55)(49,54)(50,53)(51,52)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)(93,101)(94,100)(95,99)(96,98)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109) );
G=PermutationGroup([[(1,109,82,52,40),(2,110,83,53,41),(3,111,84,54,42),(4,112,85,55,43),(5,113,86,56,44),(6,114,87,57,45),(7,115,88,58,46),(8,93,89,59,24),(9,94,90,60,25),(10,95,91,61,26),(11,96,92,62,27),(12,97,70,63,28),(13,98,71,64,29),(14,99,72,65,30),(15,100,73,66,31),(16,101,74,67,32),(17,102,75,68,33),(18,103,76,69,34),(19,104,77,47,35),(20,105,78,48,36),(21,106,79,49,37),(22,107,80,50,38),(23,108,81,51,39)], [(1,40),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(47,104),(48,105),(49,106),(50,107),(51,108),(52,109),(53,110),(54,111),(55,112),(56,113),(57,114),(58,115),(59,93),(60,94),(61,95),(62,96),(63,97),(64,98),(65,99),(66,100),(67,101),(68,102),(69,103)], [(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),(47,56),(48,55),(49,54),(50,53),(51,52),(57,69),(58,68),(59,67),(60,66),(61,65),(62,64),(71,92),(72,91),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85),(79,84),(80,83),(81,82),(93,101),(94,100),(95,99),(96,98),(102,115),(103,114),(104,113),(105,112),(106,111),(107,110),(108,109)]])
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 10A | 10B | 23A | ··· | 23K | 46A | ··· | 46K | 115A | ··· | 115V |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 10 | 23 | ··· | 23 | 46 | ··· | 46 | 115 | ··· | 115 |
| size | 1 | 5 | 23 | 115 | 2 | 2 | 46 | 46 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D5 | D10 | D23 | D46 | D5×D23 |
| kernel | D5×D23 | D5×C23 | C5×D23 | D115 | D23 | C23 | D5 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 11 | 11 | 22 |
Matrix representation of D5×D23 ►in GL4(𝔽461) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 460 |
| 0 | 0 | 24 | 438 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 460 |
| 0 | 0 | 0 | 460 |
| 304 | 1 | 0 | 0 |
| 234 | 160 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 20 | 406 | 0 | 0 |
| 309 | 441 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(461))| [1,0,0,0,0,1,0,0,0,0,1,24,0,0,460,438],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,460,460],[304,234,0,0,1,160,0,0,0,0,1,0,0,0,0,1],[20,309,0,0,406,441,0,0,0,0,1,0,0,0,0,1] >;
D5×D23 in GAP, Magma, Sage, TeX
D_5\times D_{23} % in TeX
G:=Group("D5xD23"); // GroupNames label
G:=SmallGroup(460,7);
// by ID
G=gap.SmallGroup(460,7);
# by ID
G:=PCGroup([4,-2,-2,-5,-23,102,7043]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^23=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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