metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊1D26, D104⋊2C2, C4.14D52, C52.12D4, C104⋊1C22, D52⋊4C22, C22.5D52, M4(2)⋊1D13, C52.32C23, Dic26⋊4C22, (C2×D52)⋊7C2, (C2×C26).5D4, C104⋊C2⋊1C2, C13⋊1(C8⋊C22), C26.13(C2×D4), C2.15(C2×D52), (C2×C4).15D26, D52⋊5C2⋊2C2, (C13×M4(2))⋊1C2, (C2×C52).27C22, C4.30(C22×D13), SmallGroup(416,129)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D26
G = < a,b,c | a8=b26=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >
Subgroups: 680 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C13, M4(2), D8, SD16, C2×D4, C4○D4, D13, C26, C26, C8⋊C22, Dic13, C52, D26, C2×C26, C104, Dic26, C4×D13, D52, D52, D52, C13⋊D4, C2×C52, C22×D13, C104⋊C2, D104, C13×M4(2), C2×D52, D52⋊5C2, C8⋊D26
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C8⋊C22, D26, D52, C22×D13, C2×D52, C8⋊D26
►On 104 points
Generators in S104
(1 94 52 65 23 81 27 78)(2 82 40 53 24 95 28 66)(3 96 41 67 25 83 29 54)(4 84 42 55 26 97 30 68)(5 98 43 69 14 85 31 56)(6 86 44 57 15 99 32 70)(7 100 45 71 16 87 33 58)(8 88 46 59 17 101 34 72)(9 102 47 73 18 89 35 60)(10 90 48 61 19 103 36 74)(11 104 49 75 20 91 37 62)(12 92 50 63 21 79 38 76)(13 80 51 77 22 93 39 64)
(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 4)(2 3)(5 13)(6 12)(7 11)(8 10)(14 22)(15 21)(16 20)(17 19)(23 26)(24 25)(27 42)(28 41)(29 40)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(53 83)(54 82)(55 81)(56 80)(57 79)(58 104)(59 103)(60 102)(61 101)(62 100)(63 99)(64 98)(65 97)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)
G:=sub<Sym(104)| (1,94,52,65,23,81,27,78)(2,82,40,53,24,95,28,66)(3,96,41,67,25,83,29,54)(4,84,42,55,26,97,30,68)(5,98,43,69,14,85,31,56)(6,86,44,57,15,99,32,70)(7,100,45,71,16,87,33,58)(8,88,46,59,17,101,34,72)(9,102,47,73,18,89,35,60)(10,90,48,61,19,103,36,74)(11,104,49,75,20,91,37,62)(12,92,50,63,21,79,38,76)(13,80,51,77,22,93,39,64), (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), (1,4)(2,3)(5,13)(6,12)(7,11)(8,10)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,42)(28,41)(29,40)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(53,83)(54,82)(55,81)(56,80)(57,79)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)>;
G:=Group( (1,94,52,65,23,81,27,78)(2,82,40,53,24,95,28,66)(3,96,41,67,25,83,29,54)(4,84,42,55,26,97,30,68)(5,98,43,69,14,85,31,56)(6,86,44,57,15,99,32,70)(7,100,45,71,16,87,33,58)(8,88,46,59,17,101,34,72)(9,102,47,73,18,89,35,60)(10,90,48,61,19,103,36,74)(11,104,49,75,20,91,37,62)(12,92,50,63,21,79,38,76)(13,80,51,77,22,93,39,64), (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), (1,4)(2,3)(5,13)(6,12)(7,11)(8,10)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,42)(28,41)(29,40)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(53,83)(54,82)(55,81)(56,80)(57,79)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84) );
G=PermutationGroup([[(1,94,52,65,23,81,27,78),(2,82,40,53,24,95,28,66),(3,96,41,67,25,83,29,54),(4,84,42,55,26,97,30,68),(5,98,43,69,14,85,31,56),(6,86,44,57,15,99,32,70),(7,100,45,71,16,87,33,58),(8,88,46,59,17,101,34,72),(9,102,47,73,18,89,35,60),(10,90,48,61,19,103,36,74),(11,104,49,75,20,91,37,62),(12,92,50,63,21,79,38,76),(13,80,51,77,22,93,39,64)], [(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)], [(1,4),(2,3),(5,13),(6,12),(7,11),(8,10),(14,22),(15,21),(16,20),(17,19),(23,26),(24,25),(27,42),(28,41),(29,40),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(53,83),(54,82),(55,81),(56,80),(57,79),(58,104),(59,103),(60,102),(61,101),(62,100),(63,99),(64,98),(65,97),(66,96),(67,95),(68,94),(69,93),(70,92),(71,91),(72,90),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84)]])
71 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 8A | 8B | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26L | 52A | ··· | 52L | 52M | ··· | 52R | 104A | ··· | 104X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 52 | 52 | 52 | 2 | 2 | 52 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
71 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D13 | D26 | D26 | D52 | D52 | C8⋊C22 | C8⋊D26 |
| kernel | C8⋊D26 | C104⋊C2 | D104 | C13×M4(2) | C2×D52 | D52⋊5C2 | C52 | C2×C26 | M4(2) | C8 | C2×C4 | C4 | C22 | C13 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 6 | 12 | 6 | 12 | 12 | 1 | 12 |
Matrix representation of C8⋊D26 ►in GL4(𝔽313) generated by
| 238 | 303 | 218 | 5 |
| 117 | 143 | 166 | 304 |
| 278 | 276 | 110 | 70 |
| 163 | 15 | 58 | 135 |
| 214 | 300 | 0 | 0 |
| 70 | 123 | 0 | 0 |
| 88 | 77 | 29 | 13 |
| 273 | 248 | 291 | 260 |
| 127 | 1 | 0 | 0 |
| 148 | 186 | 0 | 0 |
| 129 | 7 | 289 | 227 |
| 61 | 62 | 54 | 24 |
G:=sub<GL(4,GF(313))| [238,117,278,163,303,143,276,15,218,166,110,58,5,304,70,135],[214,70,88,273,300,123,77,248,0,0,29,291,0,0,13,260],[127,148,129,61,1,186,7,62,0,0,289,54,0,0,227,24] >;
C8⋊D26 in GAP, Magma, Sage, TeX
C_8\rtimes D_{26} % in TeX
G:=Group("C8:D26"); // GroupNames label
G:=SmallGroup(416,129);
// by ID
G=gap.SmallGroup(416,129);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,218,188,50,579,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^8=b^26=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations