direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: SD16×D13, C8⋊5D26, Q8⋊1D26, D4.2D26, C104⋊5C22, D26.24D4, C52.4C23, Dic13.8D4, D52.2C22, Dic26⋊2C22, (C8×D13)⋊4C2, Q8⋊D13⋊1C2, (Q8×D13)⋊1C2, C13⋊2(C2×SD16), C104⋊C2⋊5C2, D4.D13⋊3C2, (D4×D13).1C2, C2.18(D4×D13), C26.30(C2×D4), C13⋊2C8⋊6C22, (C13×SD16)⋊3C2, (Q8×C13)⋊1C22, C4.4(C22×D13), (D4×C13).2C22, (C4×D13).17C22, SmallGroup(416,134)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×D13
G = < a,b,c,d | a8=b2=c13=d2=1, bab=a3, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 576 in 68 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, C13, C2×C8, SD16, SD16, C2×D4, C2×Q8, D13, D13, C26, C26, C2×SD16, Dic13, Dic13, C52, C52, D26, D26, C2×C26, C13⋊2C8, C104, Dic26, Dic26, C4×D13, C4×D13, D52, C13⋊D4, D4×C13, Q8×C13, C22×D13, C8×D13, C104⋊C2, D4.D13, Q8⋊D13, C13×SD16, D4×D13, Q8×D13, SD16×D13
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, D13, C2×SD16, D26, C22×D13, D4×D13, SD16×D13
►On 104 points
Generators in S104
(1 103 38 77 25 90 51 64)(2 104 39 78 26 91 52 65)(3 92 27 66 14 79 40 53)(4 93 28 67 15 80 41 54)(5 94 29 68 16 81 42 55)(6 95 30 69 17 82 43 56)(7 96 31 70 18 83 44 57)(8 97 32 71 19 84 45 58)(9 98 33 72 20 85 46 59)(10 99 34 73 21 86 47 60)(11 100 35 74 22 87 48 61)(12 101 36 75 23 88 49 62)(13 102 37 76 24 89 50 63)
(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)
(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)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 22)(15 21)(16 20)(17 19)(23 26)(24 25)(27 35)(28 34)(29 33)(30 32)(36 39)(37 38)(40 48)(41 47)(42 46)(43 45)(49 52)(50 51)(53 61)(54 60)(55 59)(56 58)(62 65)(63 64)(66 74)(67 73)(68 72)(69 71)(75 78)(76 77)(79 87)(80 86)(81 85)(82 84)(88 91)(89 90)(92 100)(93 99)(94 98)(95 97)(101 104)(102 103)
G:=sub<Sym(104)| (1,103,38,77,25,90,51,64)(2,104,39,78,26,91,52,65)(3,92,27,66,14,79,40,53)(4,93,28,67,15,80,41,54)(5,94,29,68,16,81,42,55)(6,95,30,69,17,82,43,56)(7,96,31,70,18,83,44,57)(8,97,32,71,19,84,45,58)(9,98,33,72,20,85,46,59)(10,99,34,73,21,86,47,60)(11,100,35,74,22,87,48,61)(12,101,36,75,23,88,49,62)(13,102,37,76,24,89,50,63), (27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,35)(28,34)(29,33)(30,32)(36,39)(37,38)(40,48)(41,47)(42,46)(43,45)(49,52)(50,51)(53,61)(54,60)(55,59)(56,58)(62,65)(63,64)(66,74)(67,73)(68,72)(69,71)(75,78)(76,77)(79,87)(80,86)(81,85)(82,84)(88,91)(89,90)(92,100)(93,99)(94,98)(95,97)(101,104)(102,103)>;
G:=Group( (1,103,38,77,25,90,51,64)(2,104,39,78,26,91,52,65)(3,92,27,66,14,79,40,53)(4,93,28,67,15,80,41,54)(5,94,29,68,16,81,42,55)(6,95,30,69,17,82,43,56)(7,96,31,70,18,83,44,57)(8,97,32,71,19,84,45,58)(9,98,33,72,20,85,46,59)(10,99,34,73,21,86,47,60)(11,100,35,74,22,87,48,61)(12,101,36,75,23,88,49,62)(13,102,37,76,24,89,50,63), (27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,35)(28,34)(29,33)(30,32)(36,39)(37,38)(40,48)(41,47)(42,46)(43,45)(49,52)(50,51)(53,61)(54,60)(55,59)(56,58)(62,65)(63,64)(66,74)(67,73)(68,72)(69,71)(75,78)(76,77)(79,87)(80,86)(81,85)(82,84)(88,91)(89,90)(92,100)(93,99)(94,98)(95,97)(101,104)(102,103) );
G=PermutationGroup([[(1,103,38,77,25,90,51,64),(2,104,39,78,26,91,52,65),(3,92,27,66,14,79,40,53),(4,93,28,67,15,80,41,54),(5,94,29,68,16,81,42,55),(6,95,30,69,17,82,43,56),(7,96,31,70,18,83,44,57),(8,97,32,71,19,84,45,58),(9,98,33,72,20,85,46,59),(10,99,34,73,21,86,47,60),(11,100,35,74,22,87,48,61),(12,101,36,75,23,88,49,62),(13,102,37,76,24,89,50,63)], [(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52),(53,79),(54,80),(55,81),(56,82),(57,83),(58,84),(59,85),(60,86),(61,87),(62,88),(63,89),(64,90),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,97),(72,98),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,22),(15,21),(16,20),(17,19),(23,26),(24,25),(27,35),(28,34),(29,33),(30,32),(36,39),(37,38),(40,48),(41,47),(42,46),(43,45),(49,52),(50,51),(53,61),(54,60),(55,59),(56,58),(62,65),(63,64),(66,74),(67,73),(68,72),(69,71),(75,78),(76,77),(79,87),(80,86),(81,85),(82,84),(88,91),(89,90),(92,100),(93,99),(94,98),(95,97),(101,104),(102,103)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26L | 52A | ··· | 52F | 52G | ··· | 52L | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 4 | 13 | 13 | 52 | 2 | 4 | 26 | 52 | 2 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | SD16 | D13 | D26 | D26 | D26 | D4×D13 | SD16×D13 |
| kernel | SD16×D13 | C8×D13 | C104⋊C2 | D4.D13 | Q8⋊D13 | C13×SD16 | D4×D13 | Q8×D13 | Dic13 | D26 | D13 | SD16 | C8 | D4 | Q8 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 6 | 6 | 6 | 12 |
Matrix representation of SD16×D13 ►in GL4(𝔽313) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 183 | 130 |
| 0 | 0 | 248 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 312 |
| 115 | 1 | 0 | 0 |
| 305 | 283 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 283 | 312 | 0 | 0 |
| 273 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(313))| [1,0,0,0,0,1,0,0,0,0,183,248,0,0,130,0],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,312],[115,305,0,0,1,283,0,0,0,0,1,0,0,0,0,1],[283,273,0,0,312,30,0,0,0,0,1,0,0,0,0,1] >;
SD16×D13 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times D_{13} % in TeX
G:=Group("SD16xD13"); // GroupNames label
G:=SmallGroup(416,134);
// by ID
G=gap.SmallGroup(416,134);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,116,86,297,159,69,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^13=d^2=1,b*a*b=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations