metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C104⋊4C4, C26.2C42, D13.M4(2), C13⋊C8⋊1C4, C8⋊3(C13⋊C4), C13⋊2C8⋊8C4, C13⋊1(C8⋊C4), C52.16(C2×C4), D13⋊C8.2C2, D26.6(C2×C4), (C8×D13).10C2, Dic13.8(C2×C4), (C4×D13).32C22, (C2×C13⋊C4).C4, C2.3(C4×C13⋊C4), (C4×C13⋊C4).2C2, C4.17(C2×C13⋊C4), SmallGroup(416,67)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C104⋊C4
G = < a,b | a104=b4=1, bab-1=a5 >
Smallest permutation representation of C104⋊C4
►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)
(2 22 26 6)(3 43 51 11)(4 64 76 16)(5 85 101 21)(7 23 47 31)(8 44 72 36)(9 65 97 41)(10 86 18 46)(12 24 68 56)(13 45 93 61)(14 66)(15 87 39 71)(17 25 89 81)(19 67 35 91)(20 88 60 96)(28 48 52 32)(29 69 77 37)(30 90 102 42)(33 49 73 57)(34 70 98 62)(38 50 94 82)(40 92)(54 74 78 58)(55 95 103 63)(59 75 99 83)(80 100 104 84)
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), (2,22,26,6)(3,43,51,11)(4,64,76,16)(5,85,101,21)(7,23,47,31)(8,44,72,36)(9,65,97,41)(10,86,18,46)(12,24,68,56)(13,45,93,61)(14,66)(15,87,39,71)(17,25,89,81)(19,67,35,91)(20,88,60,96)(28,48,52,32)(29,69,77,37)(30,90,102,42)(33,49,73,57)(34,70,98,62)(38,50,94,82)(40,92)(54,74,78,58)(55,95,103,63)(59,75,99,83)(80,100,104,84)>;
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), (2,22,26,6)(3,43,51,11)(4,64,76,16)(5,85,101,21)(7,23,47,31)(8,44,72,36)(9,65,97,41)(10,86,18,46)(12,24,68,56)(13,45,93,61)(14,66)(15,87,39,71)(17,25,89,81)(19,67,35,91)(20,88,60,96)(28,48,52,32)(29,69,77,37)(30,90,102,42)(33,49,73,57)(34,70,98,62)(38,50,94,82)(40,92)(54,74,78,58)(55,95,103,63)(59,75,99,83)(80,100,104,84) );
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)], [(2,22,26,6),(3,43,51,11),(4,64,76,16),(5,85,101,21),(7,23,47,31),(8,44,72,36),(9,65,97,41),(10,86,18,46),(12,24,68,56),(13,45,93,61),(14,66),(15,87,39,71),(17,25,89,81),(19,67,35,91),(20,88,60,96),(28,48,52,32),(29,69,77,37),(30,90,102,42),(33,49,73,57),(34,70,98,62),(38,50,94,82),(40,92),(54,74,78,58),(55,95,103,63),(59,75,99,83),(80,100,104,84)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | ··· | 8H | 13A | 13B | 13C | 26A | 26B | 26C | 52A | ··· | 52F | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 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 | 1 | 1 | 13 | 13 | 26 | 26 | 26 | 26 | 2 | 2 | 26 | ··· | 26 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | M4(2) | C13⋊C4 | C2×C13⋊C4 | C4×C13⋊C4 | C104⋊C4 |
| kernel | C104⋊C4 | C8×D13 | D13⋊C8 | C4×C13⋊C4 | C13⋊2C8 | C104 | C13⋊C8 | C2×C13⋊C4 | D13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 3 | 3 | 6 | 12 |
Matrix representation of C104⋊C4 ►in GL4(𝔽5) generated by
| 2 | 3 | 0 | 3 |
| 0 | 0 | 1 | 2 |
| 3 | 0 | 3 | 3 |
| 4 | 0 | 3 | 3 |
| 1 | 1 | 3 | 0 |
| 3 | 4 | 0 | 2 |
| 3 | 3 | 1 | 3 |
| 4 | 2 | 4 | 4 |
G:=sub<GL(4,GF(5))| [2,0,3,4,3,0,0,0,0,1,3,3,3,2,3,3],[1,3,3,4,1,4,3,2,3,0,1,4,0,2,3,4] >;
C104⋊C4 in GAP, Magma, Sage, TeX
C_{104}\rtimes C_4 % in TeX
G:=Group("C104:C4"); // GroupNames label
G:=SmallGroup(416,67);
// by ID
G=gap.SmallGroup(416,67);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,217,55,69,9221,3473]);
// Polycyclic
G:=Group<a,b|a^104=b^4=1,b*a*b^-1=a^5>;
// generators/relations
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