p-group, metacyclic, nilpotent (class 3), monomial
Aliases: C8⋊2C16, C82.8C2, C8.39SD16, C4.4M5(2), (C2×C8).7C8, C4⋊C16.7C2, (C4×C8).30C4, C4.6(C2×C16), C2.3(C4⋊C16), (C2×C8).34Q8, (C2×C8).396D4, C2.2(C8⋊2C8), C4.13(C4.Q8), C2.1(C8.C8), C22.16(C4⋊C8), C42.305(C2×C4), (C4×C8).356C22, (C2×C4).56M4(2), C4.12(C8.C4), (C2×C4).71(C2×C8), (C2×C4).156(C4⋊C4), SmallGroup(128,99)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊2C16
G = < a,b | a8=b16=1, bab-1=a3 >
Smallest permutation representation of C8⋊2C16
►Regular action on 128 points
Generators in S128
(1 53 17 84 125 66 105 34)(2 85 106 54 126 35 18 67)(3 55 19 86 127 68 107 36)(4 87 108 56 128 37 20 69)(5 57 21 88 113 70 109 38)(6 89 110 58 114 39 22 71)(7 59 23 90 115 72 111 40)(8 91 112 60 116 41 24 73)(9 61 25 92 117 74 97 42)(10 93 98 62 118 43 26 75)(11 63 27 94 119 76 99 44)(12 95 100 64 120 45 28 77)(13 49 29 96 121 78 101 46)(14 81 102 50 122 47 30 79)(15 51 31 82 123 80 103 48)(16 83 104 52 124 33 32 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)
G:=sub<Sym(128)| (1,53,17,84,125,66,105,34)(2,85,106,54,126,35,18,67)(3,55,19,86,127,68,107,36)(4,87,108,56,128,37,20,69)(5,57,21,88,113,70,109,38)(6,89,110,58,114,39,22,71)(7,59,23,90,115,72,111,40)(8,91,112,60,116,41,24,73)(9,61,25,92,117,74,97,42)(10,93,98,62,118,43,26,75)(11,63,27,94,119,76,99,44)(12,95,100,64,120,45,28,77)(13,49,29,96,121,78,101,46)(14,81,102,50,122,47,30,79)(15,51,31,82,123,80,103,48)(16,83,104,52,124,33,32,65), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)>;
G:=Group( (1,53,17,84,125,66,105,34)(2,85,106,54,126,35,18,67)(3,55,19,86,127,68,107,36)(4,87,108,56,128,37,20,69)(5,57,21,88,113,70,109,38)(6,89,110,58,114,39,22,71)(7,59,23,90,115,72,111,40)(8,91,112,60,116,41,24,73)(9,61,25,92,117,74,97,42)(10,93,98,62,118,43,26,75)(11,63,27,94,119,76,99,44)(12,95,100,64,120,45,28,77)(13,49,29,96,121,78,101,46)(14,81,102,50,122,47,30,79)(15,51,31,82,123,80,103,48)(16,83,104,52,124,33,32,65), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128) );
G=PermutationGroup([[(1,53,17,84,125,66,105,34),(2,85,106,54,126,35,18,67),(3,55,19,86,127,68,107,36),(4,87,108,56,128,37,20,69),(5,57,21,88,113,70,109,38),(6,89,110,58,114,39,22,71),(7,59,23,90,115,72,111,40),(8,91,112,60,116,41,24,73),(9,61,25,92,117,74,97,42),(10,93,98,62,118,43,26,75),(11,63,27,94,119,76,99,44),(12,95,100,64,120,45,28,77),(13,49,29,96,121,78,101,46),(14,81,102,50,122,47,30,79),(15,51,31,82,123,80,103,48),(16,83,104,52,124,33,32,65)], [(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),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 8I | ··· | 8AB | 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 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C8 | C16 | D4 | Q8 | SD16 | M4(2) | C8.C4 | M5(2) | C8.C8 |
| kernel | C8⋊2C16 | C82 | C4⋊C16 | C4×C8 | C2×C8 | C8 | C2×C8 | C2×C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 1 | 1 | 4 | 2 | 4 | 4 | 8 |
Matrix representation of C8⋊2C16 ►in GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 4 | 9 |
| 0 | 14 | 0 | 0 |
| 14 | 0 | 0 | 0 |
| 0 | 0 | 7 | 15 |
| 0 | 0 | 0 | 10 |
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,15,4,0,0,0,9],[0,14,0,0,14,0,0,0,0,0,7,0,0,0,15,10] >;
C8⋊2C16 in GAP, Magma, Sage, TeX
C_8\rtimes_2C_{16} % in TeX
G:=Group("C8:2C16"); // GroupNames label
G:=SmallGroup(128,99);
// by ID
G=gap.SmallGroup(128,99);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,36,422,100,136,124]);
// Polycyclic
G:=Group<a,b|a^8=b^16=1,b*a*b^-1=a^3>;
// generators/relations
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