p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊C16, C8.18Q16, C8.38SD16, C4.2M5(2), C4⋊C4.5C8, C4.42C4≀C2, C4⋊C8.11C4, C4⋊C16.1C2, C4.2(C2×C16), (C4×C16).3C2, (C8×Q8).1C2, (C2×Q8).4C8, (C2×C8).299D4, C2.2(Q8⋊C8), (C4×Q8).10C4, C2.2(D4.C8), C2.6(C22⋊C16), C42.253(C2×C4), (C4×C8).355C22, (C2×C4).34M4(2), C4.30(Q8⋊C4), C22.36(C22⋊C8), (C2×C4).46(C2×C8), (C2×C4).382(C22⋊C4), SmallGroup(128,69)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8⋊C16
G = < a,b,c | a4=c16=1, b2=a2, bab-1=cac-1=a-1, cbc-1=a-1b >
Smallest permutation representation of Q8⋊C16
►Regular action on 128 points
Generators in S128
(1 49 37 18)(2 19 38 50)(3 51 39 20)(4 21 40 52)(5 53 41 22)(6 23 42 54)(7 55 43 24)(8 25 44 56)(9 57 45 26)(10 27 46 58)(11 59 47 28)(12 29 48 60)(13 61 33 30)(14 31 34 62)(15 63 35 32)(16 17 36 64)(65 86 107 123)(66 124 108 87)(67 88 109 125)(68 126 110 89)(69 90 111 127)(70 128 112 91)(71 92 97 113)(72 114 98 93)(73 94 99 115)(74 116 100 95)(75 96 101 117)(76 118 102 81)(77 82 103 119)(78 120 104 83)(79 84 105 121)(80 122 106 85)
(1 91 37 128)(2 71 38 97)(3 93 39 114)(4 73 40 99)(5 95 41 116)(6 75 42 101)(7 81 43 118)(8 77 44 103)(9 83 45 120)(10 79 46 105)(11 85 47 122)(12 65 48 107)(13 87 33 124)(14 67 34 109)(15 89 35 126)(16 69 36 111)(17 127 64 90)(18 70 49 112)(19 113 50 92)(20 72 51 98)(21 115 52 94)(22 74 53 100)(23 117 54 96)(24 76 55 102)(25 119 56 82)(26 78 57 104)(27 121 58 84)(28 80 59 106)(29 123 60 86)(30 66 61 108)(31 125 62 88)(32 68 63 110)
(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)| (1,49,37,18)(2,19,38,50)(3,51,39,20)(4,21,40,52)(5,53,41,22)(6,23,42,54)(7,55,43,24)(8,25,44,56)(9,57,45,26)(10,27,46,58)(11,59,47,28)(12,29,48,60)(13,61,33,30)(14,31,34,62)(15,63,35,32)(16,17,36,64)(65,86,107,123)(66,124,108,87)(67,88,109,125)(68,126,110,89)(69,90,111,127)(70,128,112,91)(71,92,97,113)(72,114,98,93)(73,94,99,115)(74,116,100,95)(75,96,101,117)(76,118,102,81)(77,82,103,119)(78,120,104,83)(79,84,105,121)(80,122,106,85), (1,91,37,128)(2,71,38,97)(3,93,39,114)(4,73,40,99)(5,95,41,116)(6,75,42,101)(7,81,43,118)(8,77,44,103)(9,83,45,120)(10,79,46,105)(11,85,47,122)(12,65,48,107)(13,87,33,124)(14,67,34,109)(15,89,35,126)(16,69,36,111)(17,127,64,90)(18,70,49,112)(19,113,50,92)(20,72,51,98)(21,115,52,94)(22,74,53,100)(23,117,54,96)(24,76,55,102)(25,119,56,82)(26,78,57,104)(27,121,58,84)(28,80,59,106)(29,123,60,86)(30,66,61,108)(31,125,62,88)(32,68,63,110), (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( (1,49,37,18)(2,19,38,50)(3,51,39,20)(4,21,40,52)(5,53,41,22)(6,23,42,54)(7,55,43,24)(8,25,44,56)(9,57,45,26)(10,27,46,58)(11,59,47,28)(12,29,48,60)(13,61,33,30)(14,31,34,62)(15,63,35,32)(16,17,36,64)(65,86,107,123)(66,124,108,87)(67,88,109,125)(68,126,110,89)(69,90,111,127)(70,128,112,91)(71,92,97,113)(72,114,98,93)(73,94,99,115)(74,116,100,95)(75,96,101,117)(76,118,102,81)(77,82,103,119)(78,120,104,83)(79,84,105,121)(80,122,106,85), (1,91,37,128)(2,71,38,97)(3,93,39,114)(4,73,40,99)(5,95,41,116)(6,75,42,101)(7,81,43,118)(8,77,44,103)(9,83,45,120)(10,79,46,105)(11,85,47,122)(12,65,48,107)(13,87,33,124)(14,67,34,109)(15,89,35,126)(16,69,36,111)(17,127,64,90)(18,70,49,112)(19,113,50,92)(20,72,51,98)(21,115,52,94)(22,74,53,100)(23,117,54,96)(24,76,55,102)(25,119,56,82)(26,78,57,104)(27,121,58,84)(28,80,59,106)(29,123,60,86)(30,66,61,108)(31,125,62,88)(32,68,63,110), (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([[(1,49,37,18),(2,19,38,50),(3,51,39,20),(4,21,40,52),(5,53,41,22),(6,23,42,54),(7,55,43,24),(8,25,44,56),(9,57,45,26),(10,27,46,58),(11,59,47,28),(12,29,48,60),(13,61,33,30),(14,31,34,62),(15,63,35,32),(16,17,36,64),(65,86,107,123),(66,124,108,87),(67,88,109,125),(68,126,110,89),(69,90,111,127),(70,128,112,91),(71,92,97,113),(72,114,98,93),(73,94,99,115),(74,116,100,95),(75,96,101,117),(76,118,102,81),(77,82,103,119),(78,120,104,83),(79,84,105,121),(80,122,106,85)], [(1,91,37,128),(2,71,38,97),(3,93,39,114),(4,73,40,99),(5,95,41,116),(6,75,42,101),(7,81,43,118),(8,77,44,103),(9,83,45,120),(10,79,46,105),(11,85,47,122),(12,65,48,107),(13,87,33,124),(14,67,34,109),(15,89,35,126),(16,69,36,111),(17,127,64,90),(18,70,49,112),(19,113,50,92),(20,72,51,98),(21,115,52,94),(22,74,53,100),(23,117,54,96),(24,76,55,102),(25,119,56,82),(26,78,57,104),(27,121,58,84),(28,80,59,106),(29,123,60,86),(30,66,61,108),(31,125,62,88),(32,68,63,110)], [(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 | 4I | 4J | 4K | 4L | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 8O | 8P | 16A | ··· | 16P | 16Q | ··· | 16X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | D4 | SD16 | Q16 | M4(2) | C4≀C2 | M5(2) | D4.C8 |
| kernel | Q8⋊C16 | C4×C16 | C4⋊C16 | C8×Q8 | C4⋊C8 | C4×Q8 | C4⋊C4 | C2×Q8 | Q8 | C2×C8 | C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of Q8⋊C16 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 16 | 0 |
| 16 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 13 |
| 11 | 0 | 0 |
| 0 | 13 | 4 |
| 0 | 4 | 4 |
G:=sub<GL(3,GF(17))| [1,0,0,0,0,16,0,1,0],[16,0,0,0,4,0,0,0,13],[11,0,0,0,13,4,0,4,4] >;
Q8⋊C16 in GAP, Magma, Sage, TeX
Q_8\rtimes C_{16} % in TeX
G:=Group("Q8:C16"); // GroupNames label
G:=SmallGroup(128,69);
// by ID
G=gap.SmallGroup(128,69);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,219,100,136,124]);
// Polycyclic
G:=Group<a,b,c|a^4=c^16=1,b^2=a^2,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^-1*b>;
// generators/relations
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