metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C104⋊2C4, D26.8D4, D13.1SD16, Dic13.2Q8, C8⋊2(C13⋊C4), C13⋊(C4.Q8), C13⋊2C8⋊5C4, C52.8(C2×C4), C26.1(C4⋊C4), (C8×D13).5C2, C52⋊C4.3C2, C2.4(C52⋊C4), (C4×D13).25C22, C4.8(C2×C13⋊C4), SmallGroup(416,68)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D26.8D4
G = < a,b,c,d | a26=b2=1, c4=a13, d2=a12b, bab=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=c3 >
Smallest permutation representation of D26.8D4
►On 104 points
Generators in S104
(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 26)(15 25)(16 24)(17 23)(18 22)(19 21)(27 41)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)(42 52)(43 51)(44 50)(45 49)(46 48)(53 61)(54 60)(55 59)(56 58)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(79 83)(80 82)(84 104)(85 103)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)
(1 101 41 64 14 88 28 77)(2 102 42 65 15 89 29 78)(3 103 43 66 16 90 30 53)(4 104 44 67 17 91 31 54)(5 79 45 68 18 92 32 55)(6 80 46 69 19 93 33 56)(7 81 47 70 20 94 34 57)(8 82 48 71 21 95 35 58)(9 83 49 72 22 96 36 59)(10 84 50 73 23 97 37 60)(11 85 51 74 24 98 38 61)(12 86 52 75 25 99 39 62)(13 87 27 76 26 100 40 63)
(1 28)(2 49 26 33)(3 44 25 38)(4 39 24 43)(5 34 23 48)(6 29 22 27)(7 50 21 32)(8 45 20 37)(9 40 19 42)(10 35 18 47)(11 30 17 52)(12 51 16 31)(13 46 15 36)(14 41)(53 54 75 74)(55 70 73 58)(56 65 72 63)(57 60 71 68)(59 76 69 78)(61 66 67 62)(64 77)(79 81 97 95)(80 102 96 100)(82 92 94 84)(83 87 93 89)(85 103 91 99)(86 98 90 104)
G:=sub<Sym(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)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(27,41)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(42,52)(43,51)(44,50)(45,49)(46,48)(53,61)(54,60)(55,59)(56,58)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(79,83)(80,82)(84,104)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95), (1,101,41,64,14,88,28,77)(2,102,42,65,15,89,29,78)(3,103,43,66,16,90,30,53)(4,104,44,67,17,91,31,54)(5,79,45,68,18,92,32,55)(6,80,46,69,19,93,33,56)(7,81,47,70,20,94,34,57)(8,82,48,71,21,95,35,58)(9,83,49,72,22,96,36,59)(10,84,50,73,23,97,37,60)(11,85,51,74,24,98,38,61)(12,86,52,75,25,99,39,62)(13,87,27,76,26,100,40,63), (1,28)(2,49,26,33)(3,44,25,38)(4,39,24,43)(5,34,23,48)(6,29,22,27)(7,50,21,32)(8,45,20,37)(9,40,19,42)(10,35,18,47)(11,30,17,52)(12,51,16,31)(13,46,15,36)(14,41)(53,54,75,74)(55,70,73,58)(56,65,72,63)(57,60,71,68)(59,76,69,78)(61,66,67,62)(64,77)(79,81,97,95)(80,102,96,100)(82,92,94,84)(83,87,93,89)(85,103,91,99)(86,98,90,104)>;
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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(27,41)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(42,52)(43,51)(44,50)(45,49)(46,48)(53,61)(54,60)(55,59)(56,58)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(79,83)(80,82)(84,104)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95), (1,101,41,64,14,88,28,77)(2,102,42,65,15,89,29,78)(3,103,43,66,16,90,30,53)(4,104,44,67,17,91,31,54)(5,79,45,68,18,92,32,55)(6,80,46,69,19,93,33,56)(7,81,47,70,20,94,34,57)(8,82,48,71,21,95,35,58)(9,83,49,72,22,96,36,59)(10,84,50,73,23,97,37,60)(11,85,51,74,24,98,38,61)(12,86,52,75,25,99,39,62)(13,87,27,76,26,100,40,63), (1,28)(2,49,26,33)(3,44,25,38)(4,39,24,43)(5,34,23,48)(6,29,22,27)(7,50,21,32)(8,45,20,37)(9,40,19,42)(10,35,18,47)(11,30,17,52)(12,51,16,31)(13,46,15,36)(14,41)(53,54,75,74)(55,70,73,58)(56,65,72,63)(57,60,71,68)(59,76,69,78)(61,66,67,62)(64,77)(79,81,97,95)(80,102,96,100)(82,92,94,84)(83,87,93,89)(85,103,91,99)(86,98,90,104) );
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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,26),(15,25),(16,24),(17,23),(18,22),(19,21),(27,41),(28,40),(29,39),(30,38),(31,37),(32,36),(33,35),(42,52),(43,51),(44,50),(45,49),(46,48),(53,61),(54,60),(55,59),(56,58),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(79,83),(80,82),(84,104),(85,103),(86,102),(87,101),(88,100),(89,99),(90,98),(91,97),(92,96),(93,95)], [(1,101,41,64,14,88,28,77),(2,102,42,65,15,89,29,78),(3,103,43,66,16,90,30,53),(4,104,44,67,17,91,31,54),(5,79,45,68,18,92,32,55),(6,80,46,69,19,93,33,56),(7,81,47,70,20,94,34,57),(8,82,48,71,21,95,35,58),(9,83,49,72,22,96,36,59),(10,84,50,73,23,97,37,60),(11,85,51,74,24,98,38,61),(12,86,52,75,25,99,39,62),(13,87,27,76,26,100,40,63)], [(1,28),(2,49,26,33),(3,44,25,38),(4,39,24,43),(5,34,23,48),(6,29,22,27),(7,50,21,32),(8,45,20,37),(9,40,19,42),(10,35,18,47),(11,30,17,52),(12,51,16,31),(13,46,15,36),(14,41),(53,54,75,74),(55,70,73,58),(56,65,72,63),(57,60,71,68),(59,76,69,78),(61,66,67,62),(64,77),(79,81,97,95),(80,102,96,100),(82,92,94,84),(83,87,93,89),(85,103,91,99),(86,98,90,104)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 13A | 13B | 13C | 26A | 26B | 26C | 52A | ··· | 52F | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | 13 | 13 | 26 | 26 | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 13 | 13 | 2 | 26 | 52 | 52 | 52 | 52 | 2 | 2 | 26 | 26 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | Q8 | D4 | SD16 | C13⋊C4 | C2×C13⋊C4 | C52⋊C4 | D26.8D4 |
| kernel | D26.8D4 | C8×D13 | C52⋊C4 | C13⋊2C8 | C104 | Dic13 | D26 | D13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 4 | 3 | 3 | 6 | 12 |
Matrix representation of D26.8D4 ►in GL4(𝔽313) generated by
| 70 | 200 | 199 | 100 |
| 213 | 84 | 84 | 213 |
| 131 | 230 | 231 | 101 |
| 243 | 213 | 243 | 0 |
| 70 | 0 | 284 | 284 |
| 213 | 1 | 101 | 101 |
| 131 | 101 | 1 | 2 |
| 243 | 212 | 242 | 241 |
| 237 | 60 | 311 | 231 |
| 2 | 179 | 2 | 0 |
| 80 | 142 | 6 | 82 |
| 251 | 251 | 0 | 257 |
| 126 | 53 | 174 | 204 |
| 190 | 260 | 160 | 160 |
| 140 | 153 | 222 | 192 |
| 27 | 0 | 283 | 18 |
G:=sub<GL(4,GF(313))| [70,213,131,243,200,84,230,213,199,84,231,243,100,213,101,0],[70,213,131,243,0,1,101,212,284,101,1,242,284,101,2,241],[237,2,80,251,60,179,142,251,311,2,6,0,231,0,82,257],[126,190,140,27,53,260,153,0,174,160,222,283,204,160,192,18] >;
D26.8D4 in GAP, Magma, Sage, TeX
D_{26}._8D_4 % in TeX
G:=Group("D26.8D4"); // GroupNames label
G:=SmallGroup(416,68);
// by ID
G=gap.SmallGroup(416,68);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,55,579,69,9221,3473]);
// Polycyclic
G:=Group<a,b,c,d|a^26=b^2=1,c^4=a^13,d^2=a^12*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^3>;
// generators/relations
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