p-group, metabelian, nilpotent (class 4), monomial
Aliases: Q16⋊3C8, C42.34D4, C8.2M4(2), C4.4C4≀C2, C8.2(C2×C8), C8⋊2C8.4C2, C2.9(D4⋊C8), (C2×C4).101D8, (C2×C8).298D4, C16⋊5C4.6C2, C2.D8.16C4, C4.4(C22⋊C8), (C4×Q16).14C2, (C2×Q16).10C4, (C2×C4).85SD16, C2.2(D8⋊2C4), (C4×C8).130C22, C2.1(C8.17D4), C22.43(D4⋊C4), (C2×C8).47(C2×C4), (C2×C4).214(C22⋊C4), SmallGroup(128,66)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q16⋊C8
G = < a,b,c | a8=c8=1, b2=a4, bab-1=a-1, cac-1=a3, cbc-1=a3b >
Smallest permutation representation of Q16⋊C8
►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 61 5 57)(2 60 6 64)(3 59 7 63)(4 58 8 62)(9 70 13 66)(10 69 14 65)(11 68 15 72)(12 67 16 71)(17 81 21 85)(18 88 22 84)(19 87 23 83)(20 86 24 82)(25 44 29 48)(26 43 30 47)(27 42 31 46)(28 41 32 45)(33 53 37 49)(34 52 38 56)(35 51 39 55)(36 50 40 54)(73 126 77 122)(74 125 78 121)(75 124 79 128)(76 123 80 127)(89 114 93 118)(90 113 94 117)(91 120 95 116)(92 119 96 115)(97 109 101 105)(98 108 102 112)(99 107 103 111)(100 106 104 110)
(1 100 34 93 31 86 14 77)(2 103 35 96 32 81 15 80)(3 98 36 91 25 84 16 75)(4 101 37 94 26 87 9 78)(5 104 38 89 27 82 10 73)(6 99 39 92 28 85 11 76)(7 102 40 95 29 88 12 79)(8 97 33 90 30 83 13 74)(17 69 127 61 107 56 115 46)(18 72 128 64 108 51 116 41)(19 67 121 59 109 54 117 44)(20 70 122 62 110 49 118 47)(21 65 123 57 111 52 119 42)(22 68 124 60 112 55 120 45)(23 71 125 63 105 50 113 48)(24 66 126 58 106 53 114 43)
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), (1,61,5,57)(2,60,6,64)(3,59,7,63)(4,58,8,62)(9,70,13,66)(10,69,14,65)(11,68,15,72)(12,67,16,71)(17,81,21,85)(18,88,22,84)(19,87,23,83)(20,86,24,82)(25,44,29,48)(26,43,30,47)(27,42,31,46)(28,41,32,45)(33,53,37,49)(34,52,38,56)(35,51,39,55)(36,50,40,54)(73,126,77,122)(74,125,78,121)(75,124,79,128)(76,123,80,127)(89,114,93,118)(90,113,94,117)(91,120,95,116)(92,119,96,115)(97,109,101,105)(98,108,102,112)(99,107,103,111)(100,106,104,110), (1,100,34,93,31,86,14,77)(2,103,35,96,32,81,15,80)(3,98,36,91,25,84,16,75)(4,101,37,94,26,87,9,78)(5,104,38,89,27,82,10,73)(6,99,39,92,28,85,11,76)(7,102,40,95,29,88,12,79)(8,97,33,90,30,83,13,74)(17,69,127,61,107,56,115,46)(18,72,128,64,108,51,116,41)(19,67,121,59,109,54,117,44)(20,70,122,62,110,49,118,47)(21,65,123,57,111,52,119,42)(22,68,124,60,112,55,120,45)(23,71,125,63,105,50,113,48)(24,66,126,58,106,53,114,43)>;
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), (1,61,5,57)(2,60,6,64)(3,59,7,63)(4,58,8,62)(9,70,13,66)(10,69,14,65)(11,68,15,72)(12,67,16,71)(17,81,21,85)(18,88,22,84)(19,87,23,83)(20,86,24,82)(25,44,29,48)(26,43,30,47)(27,42,31,46)(28,41,32,45)(33,53,37,49)(34,52,38,56)(35,51,39,55)(36,50,40,54)(73,126,77,122)(74,125,78,121)(75,124,79,128)(76,123,80,127)(89,114,93,118)(90,113,94,117)(91,120,95,116)(92,119,96,115)(97,109,101,105)(98,108,102,112)(99,107,103,111)(100,106,104,110), (1,100,34,93,31,86,14,77)(2,103,35,96,32,81,15,80)(3,98,36,91,25,84,16,75)(4,101,37,94,26,87,9,78)(5,104,38,89,27,82,10,73)(6,99,39,92,28,85,11,76)(7,102,40,95,29,88,12,79)(8,97,33,90,30,83,13,74)(17,69,127,61,107,56,115,46)(18,72,128,64,108,51,116,41)(19,67,121,59,109,54,117,44)(20,70,122,62,110,49,118,47)(21,65,123,57,111,52,119,42)(22,68,124,60,112,55,120,45)(23,71,125,63,105,50,113,48)(24,66,126,58,106,53,114,43) );
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)], [(1,61,5,57),(2,60,6,64),(3,59,7,63),(4,58,8,62),(9,70,13,66),(10,69,14,65),(11,68,15,72),(12,67,16,71),(17,81,21,85),(18,88,22,84),(19,87,23,83),(20,86,24,82),(25,44,29,48),(26,43,30,47),(27,42,31,46),(28,41,32,45),(33,53,37,49),(34,52,38,56),(35,51,39,55),(36,50,40,54),(73,126,77,122),(74,125,78,121),(75,124,79,128),(76,123,80,127),(89,114,93,118),(90,113,94,117),(91,120,95,116),(92,119,96,115),(97,109,101,105),(98,108,102,112),(99,107,103,111),(100,106,104,110)], [(1,100,34,93,31,86,14,77),(2,103,35,96,32,81,15,80),(3,98,36,91,25,84,16,75),(4,101,37,94,26,87,9,78),(5,104,38,89,27,82,10,73),(6,99,39,92,28,85,11,76),(7,102,40,95,29,88,12,79),(8,97,33,90,30,83,13,74),(17,69,127,61,107,56,115,46),(18,72,128,64,108,51,116,41),(19,67,121,59,109,54,117,44),(20,70,122,62,110,49,118,47),(21,65,123,57,111,52,119,42),(22,68,124,60,112,55,120,45),(23,71,125,63,105,50,113,48),(24,66,126,58,106,53,114,43)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D4 | M4(2) | D8 | SD16 | C4≀C2 | D8⋊2C4 | C8.17D4 |
| kernel | Q16⋊C8 | C8⋊2C8 | C16⋊5C4 | C4×Q16 | C2.D8 | C2×Q16 | Q16 | C42 | C2×C8 | C8 | C2×C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 |
Matrix representation of Q16⋊C8 ►in GL6(𝔽17)
| 12 | 13 | 0 | 0 | 0 | 0 |
| 15 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 6 |
| 0 | 0 | 14 | 3 | 6 | 11 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 14 | 6 | 3 | 11 |
| 8 | 1 | 0 | 0 | 0 | 0 |
| 5 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 2 | 1 | 2 |
| 0 | 0 | 5 | 4 | 14 | 4 |
| 0 | 0 | 5 | 13 | 3 | 6 |
| 0 | 0 | 5 | 3 | 4 | 15 |
| 10 | 9 | 0 | 0 | 0 | 0 |
| 13 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 12 | 7 | 2 |
| 0 | 0 | 4 | 2 | 8 | 16 |
| 0 | 0 | 13 | 14 | 3 | 8 |
| 0 | 0 | 9 | 10 | 16 | 4 |
G:=sub<GL(6,GF(17))| [12,15,0,0,0,0,13,5,0,0,0,0,0,0,3,14,14,14,0,0,14,3,3,6,0,0,0,6,0,3,0,0,6,11,0,11],[8,5,0,0,0,0,1,9,0,0,0,0,0,0,12,5,5,5,0,0,2,4,13,3,0,0,1,14,3,4,0,0,2,4,6,15],[10,13,0,0,0,0,9,7,0,0,0,0,0,0,8,4,13,9,0,0,12,2,14,10,0,0,7,8,3,16,0,0,2,16,8,4] >;
Q16⋊C8 in GAP, Magma, Sage, TeX
Q_{16}\rtimes C_8 % in TeX
G:=Group("Q16:C8"); // GroupNames label
G:=SmallGroup(128,66);
// by ID
G=gap.SmallGroup(128,66);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,891,100,1018,136,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c|a^8=c^8=1,b^2=a^4,b*a*b^-1=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^3*b>;
// generators/relations
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