metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D13, C104⋊4C2, D26.1C4, C4.13D26, C13⋊3M4(2), C52.13C22, Dic13.1C4, C13⋊2C8⋊4C2, C26.9(C2×C4), C2.3(C4×D13), (C4×D13).2C2, SmallGroup(208,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D13
G = < a,b,c | a8=b13=c2=1, ab=ba, cac=a5, cbc=b-1 >
Smallest permutation representation of C8⋊D13
►On 104 points
Generators in S104
(1 104 46 72 18 86 34 57)(2 92 47 73 19 87 35 58)(3 93 48 74 20 88 36 59)(4 94 49 75 21 89 37 60)(5 95 50 76 22 90 38 61)(6 96 51 77 23 91 39 62)(7 97 52 78 24 79 27 63)(8 98 40 66 25 80 28 64)(9 99 41 67 26 81 29 65)(10 100 42 68 14 82 30 53)(11 101 43 69 15 83 31 54)(12 102 44 70 16 84 32 55)(13 103 45 71 17 85 33 56)
(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 21)(15 20)(16 19)(17 18)(22 26)(23 25)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 78)(64 77)(65 76)(79 97)(80 96)(81 95)(82 94)(83 93)(84 92)(85 104)(86 103)(87 102)(88 101)(89 100)(90 99)(91 98)
G:=sub<Sym(104)| (1,104,46,72,18,86,34,57)(2,92,47,73,19,87,35,58)(3,93,48,74,20,88,36,59)(4,94,49,75,21,89,37,60)(5,95,50,76,22,90,38,61)(6,96,51,77,23,91,39,62)(7,97,52,78,24,79,27,63)(8,98,40,66,25,80,28,64)(9,99,41,67,26,81,29,65)(10,100,42,68,14,82,30,53)(11,101,43,69,15,83,31,54)(12,102,44,70,16,84,32,55)(13,103,45,71,17,85,33,56), (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)(14,21)(15,20)(16,19)(17,18)(22,26)(23,25)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,78)(64,77)(65,76)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)>;
G:=Group( (1,104,46,72,18,86,34,57)(2,92,47,73,19,87,35,58)(3,93,48,74,20,88,36,59)(4,94,49,75,21,89,37,60)(5,95,50,76,22,90,38,61)(6,96,51,77,23,91,39,62)(7,97,52,78,24,79,27,63)(8,98,40,66,25,80,28,64)(9,99,41,67,26,81,29,65)(10,100,42,68,14,82,30,53)(11,101,43,69,15,83,31,54)(12,102,44,70,16,84,32,55)(13,103,45,71,17,85,33,56), (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)(14,21)(15,20)(16,19)(17,18)(22,26)(23,25)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,78)(64,77)(65,76)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98) );
G=PermutationGroup([[(1,104,46,72,18,86,34,57),(2,92,47,73,19,87,35,58),(3,93,48,74,20,88,36,59),(4,94,49,75,21,89,37,60),(5,95,50,76,22,90,38,61),(6,96,51,77,23,91,39,62),(7,97,52,78,24,79,27,63),(8,98,40,66,25,80,28,64),(9,99,41,67,26,81,29,65),(10,100,42,68,14,82,30,53),(11,101,43,69,15,83,31,54),(12,102,44,70,16,84,32,55),(13,103,45,71,17,85,33,56)], [(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),(14,21),(15,20),(16,19),(17,18),(22,26),(23,25),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(53,75),(54,74),(55,73),(56,72),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,78),(64,77),(65,76),(79,97),(80,96),(81,95),(82,94),(83,93),(84,92),(85,104),(86,103),(87,102),(88,101),(89,100),(90,99),(91,98)]])
C8⋊D13 is a maximal subgroup of
D52.3C4 M4(2)×D13 D52.2C4 D8⋊D13 Q8⋊D26 D4.D26 Q16⋊D13
C8⋊D13 is a maximal quotient of C52.8Q8 C104⋊8C4 D26⋊1C8
58 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26F | 52A | ··· | 52L | 104A | ··· | 104X |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 26 | 1 | 1 | 26 | 2 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | M4(2) | D13 | D26 | C4×D13 | C8⋊D13 |
| kernel | C8⋊D13 | C13⋊2C8 | C104 | C4×D13 | Dic13 | D26 | C13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 12 | 24 |
Matrix representation of C8⋊D13 ►in GL2(𝔽313) generated by
| 205 | 71 |
| 8 | 108 |
| 52 | 1 |
| 150 | 33 |
| 33 | 312 |
| 149 | 280 |
G:=sub<GL(2,GF(313))| [205,8,71,108],[52,150,1,33],[33,149,312,280] >;
C8⋊D13 in GAP, Magma, Sage, TeX
C_8\rtimes D_{13} % in TeX
G:=Group("C8:D13"); // GroupNames label
G:=SmallGroup(208,5);
// by ID
G=gap.SmallGroup(208,5);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,101,26,42,4804]);
// Polycyclic
G:=Group<a,b,c|a^8=b^13=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations
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