p-group, metacyclic, nilpotent (class 4), monomial
Aliases: C16⋊3C8, C4.9Q32, C4.17D16, C42.311D4, C8.14M4(2), C4.5(C4⋊C8), (C4×C16).7C2, C8.14(C2×C8), (C2×C8).36Q8, C8⋊1C8.5C2, (C2×C16).12C4, (C2×C4).160D8, (C2×C4).27Q16, C2.3(C8⋊1C8), C2.1(C16⋊3C4), C4.3(C8.C4), (C4×C8).386C22, C2.1(C8.4Q8), C22.16(C2.D8), (C2×C8).216(C2×C4), (C2×C4).101(C4⋊C4), SmallGroup(128,103)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16⋊3C8
G = < a,b | a16=b8=1, bab-1=a-1 >
Smallest permutation representation of C16⋊3C8
►Regular action on 128 points
Generators in S128
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)
(1 128 93 98 38 24 72 60)(2 127 94 97 39 23 73 59)(3 126 95 112 40 22 74 58)(4 125 96 111 41 21 75 57)(5 124 81 110 42 20 76 56)(6 123 82 109 43 19 77 55)(7 122 83 108 44 18 78 54)(8 121 84 107 45 17 79 53)(9 120 85 106 46 32 80 52)(10 119 86 105 47 31 65 51)(11 118 87 104 48 30 66 50)(12 117 88 103 33 29 67 49)(13 116 89 102 34 28 68 64)(14 115 90 101 35 27 69 63)(15 114 91 100 36 26 70 62)(16 113 92 99 37 25 71 61)
G:=sub<Sym(128)| (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128), (1,128,93,98,38,24,72,60)(2,127,94,97,39,23,73,59)(3,126,95,112,40,22,74,58)(4,125,96,111,41,21,75,57)(5,124,81,110,42,20,76,56)(6,123,82,109,43,19,77,55)(7,122,83,108,44,18,78,54)(8,121,84,107,45,17,79,53)(9,120,85,106,46,32,80,52)(10,119,86,105,47,31,65,51)(11,118,87,104,48,30,66,50)(12,117,88,103,33,29,67,49)(13,116,89,102,34,28,68,64)(14,115,90,101,35,27,69,63)(15,114,91,100,36,26,70,62)(16,113,92,99,37,25,71,61)>;
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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128), (1,128,93,98,38,24,72,60)(2,127,94,97,39,23,73,59)(3,126,95,112,40,22,74,58)(4,125,96,111,41,21,75,57)(5,124,81,110,42,20,76,56)(6,123,82,109,43,19,77,55)(7,122,83,108,44,18,78,54)(8,121,84,107,45,17,79,53)(9,120,85,106,46,32,80,52)(10,119,86,105,47,31,65,51)(11,118,87,104,48,30,66,50)(12,117,88,103,33,29,67,49)(13,116,89,102,34,28,68,64)(14,115,90,101,35,27,69,63)(15,114,91,100,36,26,70,62)(16,113,92,99,37,25,71,61) );
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),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)], [(1,128,93,98,38,24,72,60),(2,127,94,97,39,23,73,59),(3,126,95,112,40,22,74,58),(4,125,96,111,41,21,75,57),(5,124,81,110,42,20,76,56),(6,123,82,109,43,19,77,55),(7,122,83,108,44,18,78,54),(8,121,84,107,45,17,79,53),(9,120,85,106,46,32,80,52),(10,119,86,105,47,31,65,51),(11,118,87,104,48,30,66,50),(12,117,88,103,33,29,67,49),(13,116,89,102,34,28,68,64),(14,115,90,101,35,27,69,63),(15,114,91,100,36,26,70,62),(16,113,92,99,37,25,71,61)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 8I | ··· | 8P | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | |||||
| image | C1 | C2 | C2 | C4 | C8 | D4 | Q8 | M4(2) | D8 | Q16 | C8.C4 | D16 | Q32 | C8.4Q8 |
| kernel | C16⋊3C8 | C8⋊1C8 | C4×C16 | C2×C16 | C16 | C42 | C2×C8 | C8 | C2×C4 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C16⋊3C8 ►in GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 9 | 12 |
| 3 | 2 | 0 | 0 |
| 2 | 14 | 0 | 0 |
| 0 | 0 | 8 | 15 |
| 0 | 0 | 7 | 9 |
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,10,9,0,0,0,12],[3,2,0,0,2,14,0,0,0,0,8,7,0,0,15,9] >;
C16⋊3C8 in GAP, Magma, Sage, TeX
C_{16}\rtimes_3C_8 % in TeX
G:=Group("C16:3C8"); // GroupNames label
G:=SmallGroup(128,103);
// by ID
G=gap.SmallGroup(128,103);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,148,422,436,136,2804,172]);
// Polycyclic
G:=Group<a,b|a^16=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export