direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×C13⋊C4, C104⋊3C4, C26.1C42, C13⋊C8⋊3C4, C13⋊1(C4×C8), D13.(C2×C8), C13⋊2C8⋊7C4, C52.15(C2×C4), D13⋊C8.3C2, (C8×D13).9C2, D26.5(C2×C4), Dic13.7(C2×C4), (C4×D13).31C22, C2.1(C4×C13⋊C4), (C4×C13⋊C4).3C2, (C2×C13⋊C4).2C4, C4.16(C2×C13⋊C4), SmallGroup(416,66)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C8×C13⋊C4 |
Generators and relations for C8×C13⋊C4
G = < a,b,c | a8=b13=c4=1, ab=ba, ac=ca, cbc-1=b5 >
13C2
13C2
13C4
13C4
13C22
13C4
13C4
13C4
13C8
13C8
13C2×C4
13C2×C4
13C2×C4
13C8
13C2×C8
13C42
13C2×C8
13C4×C8
Smallest permutation representation of C8×C13⋊C4
►On 104 points
Generators in S104
(1 92 40 66 14 79 27 53)(2 93 41 67 15 80 28 54)(3 94 42 68 16 81 29 55)(4 95 43 69 17 82 30 56)(5 96 44 70 18 83 31 57)(6 97 45 71 19 84 32 58)(7 98 46 72 20 85 33 59)(8 99 47 73 21 86 34 60)(9 100 48 74 22 87 35 61)(10 101 49 75 23 88 36 62)(11 102 50 76 24 89 37 63)(12 103 51 77 25 90 38 64)(13 104 52 78 26 91 39 65)
(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 14)(2 22 13 19)(3 17 12 24)(4 25 11 16)(5 20 10 21)(6 15 9 26)(7 23 8 18)(27 40)(28 48 39 45)(29 43 38 50)(30 51 37 42)(31 46 36 47)(32 41 35 52)(33 49 34 44)(53 66)(54 74 65 71)(55 69 64 76)(56 77 63 68)(57 72 62 73)(58 67 61 78)(59 75 60 70)(79 92)(80 100 91 97)(81 95 90 102)(82 103 89 94)(83 98 88 99)(84 93 87 104)(85 101 86 96)
G:=sub<Sym(104)| (1,92,40,66,14,79,27,53)(2,93,41,67,15,80,28,54)(3,94,42,68,16,81,29,55)(4,95,43,69,17,82,30,56)(5,96,44,70,18,83,31,57)(6,97,45,71,19,84,32,58)(7,98,46,72,20,85,33,59)(8,99,47,73,21,86,34,60)(9,100,48,74,22,87,35,61)(10,101,49,75,23,88,36,62)(11,102,50,76,24,89,37,63)(12,103,51,77,25,90,38,64)(13,104,52,78,26,91,39,65), (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), (1,14)(2,22,13,19)(3,17,12,24)(4,25,11,16)(5,20,10,21)(6,15,9,26)(7,23,8,18)(27,40)(28,48,39,45)(29,43,38,50)(30,51,37,42)(31,46,36,47)(32,41,35,52)(33,49,34,44)(53,66)(54,74,65,71)(55,69,64,76)(56,77,63,68)(57,72,62,73)(58,67,61,78)(59,75,60,70)(79,92)(80,100,91,97)(81,95,90,102)(82,103,89,94)(83,98,88,99)(84,93,87,104)(85,101,86,96)>;
G:=Group( (1,92,40,66,14,79,27,53)(2,93,41,67,15,80,28,54)(3,94,42,68,16,81,29,55)(4,95,43,69,17,82,30,56)(5,96,44,70,18,83,31,57)(6,97,45,71,19,84,32,58)(7,98,46,72,20,85,33,59)(8,99,47,73,21,86,34,60)(9,100,48,74,22,87,35,61)(10,101,49,75,23,88,36,62)(11,102,50,76,24,89,37,63)(12,103,51,77,25,90,38,64)(13,104,52,78,26,91,39,65), (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), (1,14)(2,22,13,19)(3,17,12,24)(4,25,11,16)(5,20,10,21)(6,15,9,26)(7,23,8,18)(27,40)(28,48,39,45)(29,43,38,50)(30,51,37,42)(31,46,36,47)(32,41,35,52)(33,49,34,44)(53,66)(54,74,65,71)(55,69,64,76)(56,77,63,68)(57,72,62,73)(58,67,61,78)(59,75,60,70)(79,92)(80,100,91,97)(81,95,90,102)(82,103,89,94)(83,98,88,99)(84,93,87,104)(85,101,86,96) );
G=PermutationGroup([[(1,92,40,66,14,79,27,53),(2,93,41,67,15,80,28,54),(3,94,42,68,16,81,29,55),(4,95,43,69,17,82,30,56),(5,96,44,70,18,83,31,57),(6,97,45,71,19,84,32,58),(7,98,46,72,20,85,33,59),(8,99,47,73,21,86,34,60),(9,100,48,74,22,87,35,61),(10,101,49,75,23,88,36,62),(11,102,50,76,24,89,37,63),(12,103,51,77,25,90,38,64),(13,104,52,78,26,91,39,65)], [(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)], [(1,14),(2,22,13,19),(3,17,12,24),(4,25,11,16),(5,20,10,21),(6,15,9,26),(7,23,8,18),(27,40),(28,48,39,45),(29,43,38,50),(30,51,37,42),(31,46,36,47),(32,41,35,52),(33,49,34,44),(53,66),(54,74,65,71),(55,69,64,76),(56,77,63,68),(57,72,62,73),(58,67,61,78),(59,75,60,70),(79,92),(80,100,91,97),(81,95,90,102),(82,103,89,94),(83,98,88,99),(84,93,87,104),(85,101,86,96)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | ··· | 4L | 8A | 8B | 8C | 8D | 8E | ··· | 8P | 13A | 13B | 13C | 26A | 26B | 26C | 52A | ··· | 52F | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 13 | 13 | 13 | 26 | 26 | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 13 | 13 | 1 | 1 | 13 | ··· | 13 | 1 | 1 | 1 | 1 | 13 | ··· | 13 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | C13⋊C4 | C2×C13⋊C4 | C4×C13⋊C4 | C8×C13⋊C4 |
| kernel | C8×C13⋊C4 | C8×D13 | D13⋊C8 | C4×C13⋊C4 | C13⋊2C8 | C104 | C13⋊C8 | C2×C13⋊C4 | C13⋊C4 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 3 | 3 | 6 | 12 |
Matrix representation of C8×C13⋊C4 ►in GL4(𝔽313) generated by
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 312 | 312 | 312 | 28 |
| 1 | 0 | 0 | 213 |
| 0 | 1 | 0 | 100 |
| 0 | 0 | 1 | 284 |
| 101 | 0 | 183 | 214 |
| 285 | 0 | 201 | 201 |
| 100 | 312 | 214 | 183 |
| 211 | 0 | 29 | 311 |
G:=sub<GL(4,GF(313))| [5,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[312,1,0,0,312,0,1,0,312,0,0,1,28,213,100,284],[101,285,100,211,0,0,312,0,183,201,214,29,214,201,183,311] >;
C8×C13⋊C4 in GAP, Magma, Sage, TeX
C_8\times C_{13}\rtimes C_4 % in TeX
G:=Group("C8xC13:C4"); // GroupNames label
G:=SmallGroup(416,66);
// by ID
G=gap.SmallGroup(416,66);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,55,69,9221,3473]);
// Polycyclic
G:=Group<a,b,c|a^8=b^13=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
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