p-group, metabelian, nilpotent (class 4), monomial
Aliases: Q16⋊1C8, C4.8Q32, C4.15SD32, C8.7M4(2), C42.310D4, C4.2C4≀C2, C8.8(C2×C8), (C4×C16).2C2, C8⋊1C8.1C2, C2.D8.7C4, C2.7(D4⋊C8), (C2×C8).296D4, (C2×C4).159D8, (C4×Q16).1C2, (C2×Q16).5C4, C4.2(C22⋊C8), (C2×C4).60SD16, (C4×C8).385C22, C2.1(C2.Q32), C2.2(D8.C4), C22.41(D4⋊C4), (C2×C8).166(C2×C4), (C2×C4).212(C22⋊C4), SmallGroup(128,64)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q16⋊1C8
G = < a,b,c | a8=c8=1, b2=a4, bab-1=cac-1=a-1, cbc-1=a-1b >
Smallest permutation representation of Q16⋊1C8
►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 57 5 61)(2 64 6 60)(3 63 7 59)(4 62 8 58)(9 70 13 66)(10 69 14 65)(11 68 15 72)(12 67 16 71)(17 86 21 82)(18 85 22 81)(19 84 23 88)(20 83 24 87)(25 41 29 45)(26 48 30 44)(27 47 31 43)(28 46 32 42)(33 50 37 54)(34 49 38 53)(35 56 39 52)(36 55 40 51)(73 128 77 124)(74 127 78 123)(75 126 79 122)(76 125 80 121)(89 115 93 119)(90 114 94 118)(91 113 95 117)(92 120 96 116)(97 105 101 109)(98 112 102 108)(99 111 103 107)(100 110 104 106)
(1 101 39 93 27 88 10 75)(2 100 40 92 28 87 11 74)(3 99 33 91 29 86 12 73)(4 98 34 90 30 85 13 80)(5 97 35 89 31 84 14 79)(6 104 36 96 32 83 15 78)(7 103 37 95 25 82 16 77)(8 102 38 94 26 81 9 76)(17 72 124 60 107 55 117 42)(18 71 125 59 108 54 118 41)(19 70 126 58 109 53 119 48)(20 69 127 57 110 52 120 47)(21 68 128 64 111 51 113 46)(22 67 121 63 112 50 114 45)(23 66 122 62 105 49 115 44)(24 65 123 61 106 56 116 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), (1,57,5,61)(2,64,6,60)(3,63,7,59)(4,62,8,58)(9,70,13,66)(10,69,14,65)(11,68,15,72)(12,67,16,71)(17,86,21,82)(18,85,22,81)(19,84,23,88)(20,83,24,87)(25,41,29,45)(26,48,30,44)(27,47,31,43)(28,46,32,42)(33,50,37,54)(34,49,38,53)(35,56,39,52)(36,55,40,51)(73,128,77,124)(74,127,78,123)(75,126,79,122)(76,125,80,121)(89,115,93,119)(90,114,94,118)(91,113,95,117)(92,120,96,116)(97,105,101,109)(98,112,102,108)(99,111,103,107)(100,110,104,106), (1,101,39,93,27,88,10,75)(2,100,40,92,28,87,11,74)(3,99,33,91,29,86,12,73)(4,98,34,90,30,85,13,80)(5,97,35,89,31,84,14,79)(6,104,36,96,32,83,15,78)(7,103,37,95,25,82,16,77)(8,102,38,94,26,81,9,76)(17,72,124,60,107,55,117,42)(18,71,125,59,108,54,118,41)(19,70,126,58,109,53,119,48)(20,69,127,57,110,52,120,47)(21,68,128,64,111,51,113,46)(22,67,121,63,112,50,114,45)(23,66,122,62,105,49,115,44)(24,65,123,61,106,56,116,43)>;
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), (1,57,5,61)(2,64,6,60)(3,63,7,59)(4,62,8,58)(9,70,13,66)(10,69,14,65)(11,68,15,72)(12,67,16,71)(17,86,21,82)(18,85,22,81)(19,84,23,88)(20,83,24,87)(25,41,29,45)(26,48,30,44)(27,47,31,43)(28,46,32,42)(33,50,37,54)(34,49,38,53)(35,56,39,52)(36,55,40,51)(73,128,77,124)(74,127,78,123)(75,126,79,122)(76,125,80,121)(89,115,93,119)(90,114,94,118)(91,113,95,117)(92,120,96,116)(97,105,101,109)(98,112,102,108)(99,111,103,107)(100,110,104,106), (1,101,39,93,27,88,10,75)(2,100,40,92,28,87,11,74)(3,99,33,91,29,86,12,73)(4,98,34,90,30,85,13,80)(5,97,35,89,31,84,14,79)(6,104,36,96,32,83,15,78)(7,103,37,95,25,82,16,77)(8,102,38,94,26,81,9,76)(17,72,124,60,107,55,117,42)(18,71,125,59,108,54,118,41)(19,70,126,58,109,53,119,48)(20,69,127,57,110,52,120,47)(21,68,128,64,111,51,113,46)(22,67,121,63,112,50,114,45)(23,66,122,62,105,49,115,44)(24,65,123,61,106,56,116,43) );
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)], [(1,57,5,61),(2,64,6,60),(3,63,7,59),(4,62,8,58),(9,70,13,66),(10,69,14,65),(11,68,15,72),(12,67,16,71),(17,86,21,82),(18,85,22,81),(19,84,23,88),(20,83,24,87),(25,41,29,45),(26,48,30,44),(27,47,31,43),(28,46,32,42),(33,50,37,54),(34,49,38,53),(35,56,39,52),(36,55,40,51),(73,128,77,124),(74,127,78,123),(75,126,79,122),(76,125,80,121),(89,115,93,119),(90,114,94,118),(91,113,95,117),(92,120,96,116),(97,105,101,109),(98,112,102,108),(99,111,103,107),(100,110,104,106)], [(1,101,39,93,27,88,10,75),(2,100,40,92,28,87,11,74),(3,99,33,91,29,86,12,73),(4,98,34,90,30,85,13,80),(5,97,35,89,31,84,14,79),(6,104,36,96,32,83,15,78),(7,103,37,95,25,82,16,77),(8,102,38,94,26,81,9,76),(17,72,124,60,107,55,117,42),(18,71,125,59,108,54,118,41),(19,70,126,58,109,53,119,48),(20,69,127,57,110,52,120,47),(21,68,128,64,111,51,113,46),(22,67,121,63,112,50,114,45),(23,66,122,62,105,49,115,44),(24,65,123,61,106,56,116,43)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D4 | M4(2) | D8 | SD16 | C4≀C2 | SD32 | Q32 | D8.C4 |
| kernel | Q16⋊1C8 | C8⋊1C8 | C4×C16 | C4×Q16 | C2.D8 | C2×Q16 | Q16 | C42 | C2×C8 | C8 | C2×C4 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of Q16⋊1C8 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 14 | 3 |
| 0 | 14 | 14 |
| 1 | 0 | 0 |
| 0 | 0 | 13 |
| 0 | 13 | 0 |
| 9 | 0 | 0 |
| 0 | 7 | 1 |
| 0 | 1 | 10 |
G:=sub<GL(3,GF(17))| [1,0,0,0,14,14,0,3,14],[1,0,0,0,0,13,0,13,0],[9,0,0,0,7,1,0,1,10] >;
Q16⋊1C8 in GAP, Magma, Sage, TeX
Q_{16}\rtimes_1C_8 % in TeX
G:=Group("Q16:1C8"); // GroupNames label
G:=SmallGroup(128,64);
// by ID
G=gap.SmallGroup(128,64);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,219,436,136,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c|a^8=c^8=1,b^2=a^4,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^-1*b>;
// generators/relations
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