p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q16⋊5Q8, C42.71C23, C4.1052- 1+4, C8⋊Q8.3C2, C8.10(C2×Q8), C2.48(D4×Q8), C4⋊C4.393D4, Q8.14(C2×Q8), C8⋊4Q8.7C2, Q8.Q8.4C2, (C4×Q16).18C2, Q8⋊3Q8.8C2, (C2×Q8).142D4, C2.69(Q8○D8), C8.5Q8.8C2, C4.48(C22×Q8), C4⋊C4.279C23, C4⋊C8.149C22, (C2×C8).217C23, (C2×C4).582C24, (C4×C8).206C22, Q16⋊C4.3C2, C4.Q16.12C2, C4⋊Q8.211C22, C2.D8.77C22, C4.Q8.80C22, C8⋊C4.75C22, (C2×Q8).415C23, (C4×Q8).209C22, (C2×Q16).165C22, Q8⋊C4.93C22, C22.842(C22×D4), C42.C2.80C22, C2.107(D8⋊C22), (C2×C4).652(C2×D4), SmallGroup(128,2122)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q16⋊5Q8
G = < a,b,c,d | a8=c4=1, b2=a4, d2=c2, bab-1=a-1, ac=ca, dad-1=a5, cbc-1=dbd-1=a4b, dcd-1=c-1 >
Subgroups: 248 in 160 conjugacy classes, 94 normal (26 characteristic)
C1, C2, C4, C4, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C42, C42, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, C2×Q8, C4×C8, C8⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C2.D8, C4×Q8, C4×Q8, C4×Q8, C42.C2, C42.C2, C4⋊Q8, C4⋊Q8, C2×Q16, C4×Q16, Q16⋊C4, C8⋊4Q8, C4.Q16, Q8.Q8, C8.5Q8, C8⋊Q8, Q8⋊3Q8, Q16⋊5Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C22×D4, C22×Q8, 2- 1+4, D4×Q8, D8⋊C22, Q8○D8, Q16⋊5Q8
Character table of Q16⋊5Q8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ15 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | -2 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ27 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ28 | 4 | -4 | -4 | 4 | 4i | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ29 | 4 | -4 | -4 | 4 | -4i | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
►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 51 5 55)(2 50 6 54)(3 49 7 53)(4 56 8 52)(9 128 13 124)(10 127 14 123)(11 126 15 122)(12 125 16 121)(17 107 21 111)(18 106 22 110)(19 105 23 109)(20 112 24 108)(25 104 29 100)(26 103 30 99)(27 102 31 98)(28 101 32 97)(33 59 37 63)(34 58 38 62)(35 57 39 61)(36 64 40 60)(41 70 45 66)(42 69 46 65)(43 68 47 72)(44 67 48 71)(73 118 77 114)(74 117 78 113)(75 116 79 120)(76 115 80 119)(81 91 85 95)(82 90 86 94)(83 89 87 93)(84 96 88 92)
(1 32 43 105)(2 25 44 106)(3 26 45 107)(4 27 46 108)(5 28 47 109)(6 29 48 110)(7 30 41 111)(8 31 42 112)(9 83 114 39)(10 84 115 40)(11 85 116 33)(12 86 117 34)(13 87 118 35)(14 88 119 36)(15 81 120 37)(16 82 113 38)(17 49 99 66)(18 50 100 67)(19 51 101 68)(20 52 102 69)(21 53 103 70)(22 54 104 71)(23 55 97 72)(24 56 98 65)(57 128 93 73)(58 121 94 74)(59 122 95 75)(60 123 96 76)(61 124 89 77)(62 125 90 78)(63 126 91 79)(64 127 92 80)
(1 62 43 90)(2 59 44 95)(3 64 45 92)(4 61 46 89)(5 58 47 94)(6 63 48 91)(7 60 41 96)(8 57 42 93)(9 98 114 24)(10 103 115 21)(11 100 116 18)(12 97 117 23)(13 102 118 20)(14 99 119 17)(15 104 120 22)(16 101 113 19)(25 75 106 122)(26 80 107 127)(27 77 108 124)(28 74 109 121)(29 79 110 126)(30 76 111 123)(31 73 112 128)(32 78 105 125)(33 67 85 50)(34 72 86 55)(35 69 87 52)(36 66 88 49)(37 71 81 54)(38 68 82 51)(39 65 83 56)(40 70 84 53)
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), (1,51,5,55)(2,50,6,54)(3,49,7,53)(4,56,8,52)(9,128,13,124)(10,127,14,123)(11,126,15,122)(12,125,16,121)(17,107,21,111)(18,106,22,110)(19,105,23,109)(20,112,24,108)(25,104,29,100)(26,103,30,99)(27,102,31,98)(28,101,32,97)(33,59,37,63)(34,58,38,62)(35,57,39,61)(36,64,40,60)(41,70,45,66)(42,69,46,65)(43,68,47,72)(44,67,48,71)(73,118,77,114)(74,117,78,113)(75,116,79,120)(76,115,80,119)(81,91,85,95)(82,90,86,94)(83,89,87,93)(84,96,88,92), (1,32,43,105)(2,25,44,106)(3,26,45,107)(4,27,46,108)(5,28,47,109)(6,29,48,110)(7,30,41,111)(8,31,42,112)(9,83,114,39)(10,84,115,40)(11,85,116,33)(12,86,117,34)(13,87,118,35)(14,88,119,36)(15,81,120,37)(16,82,113,38)(17,49,99,66)(18,50,100,67)(19,51,101,68)(20,52,102,69)(21,53,103,70)(22,54,104,71)(23,55,97,72)(24,56,98,65)(57,128,93,73)(58,121,94,74)(59,122,95,75)(60,123,96,76)(61,124,89,77)(62,125,90,78)(63,126,91,79)(64,127,92,80), (1,62,43,90)(2,59,44,95)(3,64,45,92)(4,61,46,89)(5,58,47,94)(6,63,48,91)(7,60,41,96)(8,57,42,93)(9,98,114,24)(10,103,115,21)(11,100,116,18)(12,97,117,23)(13,102,118,20)(14,99,119,17)(15,104,120,22)(16,101,113,19)(25,75,106,122)(26,80,107,127)(27,77,108,124)(28,74,109,121)(29,79,110,126)(30,76,111,123)(31,73,112,128)(32,78,105,125)(33,67,85,50)(34,72,86,55)(35,69,87,52)(36,66,88,49)(37,71,81,54)(38,68,82,51)(39,65,83,56)(40,70,84,53)>;
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), (1,51,5,55)(2,50,6,54)(3,49,7,53)(4,56,8,52)(9,128,13,124)(10,127,14,123)(11,126,15,122)(12,125,16,121)(17,107,21,111)(18,106,22,110)(19,105,23,109)(20,112,24,108)(25,104,29,100)(26,103,30,99)(27,102,31,98)(28,101,32,97)(33,59,37,63)(34,58,38,62)(35,57,39,61)(36,64,40,60)(41,70,45,66)(42,69,46,65)(43,68,47,72)(44,67,48,71)(73,118,77,114)(74,117,78,113)(75,116,79,120)(76,115,80,119)(81,91,85,95)(82,90,86,94)(83,89,87,93)(84,96,88,92), (1,32,43,105)(2,25,44,106)(3,26,45,107)(4,27,46,108)(5,28,47,109)(6,29,48,110)(7,30,41,111)(8,31,42,112)(9,83,114,39)(10,84,115,40)(11,85,116,33)(12,86,117,34)(13,87,118,35)(14,88,119,36)(15,81,120,37)(16,82,113,38)(17,49,99,66)(18,50,100,67)(19,51,101,68)(20,52,102,69)(21,53,103,70)(22,54,104,71)(23,55,97,72)(24,56,98,65)(57,128,93,73)(58,121,94,74)(59,122,95,75)(60,123,96,76)(61,124,89,77)(62,125,90,78)(63,126,91,79)(64,127,92,80), (1,62,43,90)(2,59,44,95)(3,64,45,92)(4,61,46,89)(5,58,47,94)(6,63,48,91)(7,60,41,96)(8,57,42,93)(9,98,114,24)(10,103,115,21)(11,100,116,18)(12,97,117,23)(13,102,118,20)(14,99,119,17)(15,104,120,22)(16,101,113,19)(25,75,106,122)(26,80,107,127)(27,77,108,124)(28,74,109,121)(29,79,110,126)(30,76,111,123)(31,73,112,128)(32,78,105,125)(33,67,85,50)(34,72,86,55)(35,69,87,52)(36,66,88,49)(37,71,81,54)(38,68,82,51)(39,65,83,56)(40,70,84,53) );
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)], [(1,51,5,55),(2,50,6,54),(3,49,7,53),(4,56,8,52),(9,128,13,124),(10,127,14,123),(11,126,15,122),(12,125,16,121),(17,107,21,111),(18,106,22,110),(19,105,23,109),(20,112,24,108),(25,104,29,100),(26,103,30,99),(27,102,31,98),(28,101,32,97),(33,59,37,63),(34,58,38,62),(35,57,39,61),(36,64,40,60),(41,70,45,66),(42,69,46,65),(43,68,47,72),(44,67,48,71),(73,118,77,114),(74,117,78,113),(75,116,79,120),(76,115,80,119),(81,91,85,95),(82,90,86,94),(83,89,87,93),(84,96,88,92)], [(1,32,43,105),(2,25,44,106),(3,26,45,107),(4,27,46,108),(5,28,47,109),(6,29,48,110),(7,30,41,111),(8,31,42,112),(9,83,114,39),(10,84,115,40),(11,85,116,33),(12,86,117,34),(13,87,118,35),(14,88,119,36),(15,81,120,37),(16,82,113,38),(17,49,99,66),(18,50,100,67),(19,51,101,68),(20,52,102,69),(21,53,103,70),(22,54,104,71),(23,55,97,72),(24,56,98,65),(57,128,93,73),(58,121,94,74),(59,122,95,75),(60,123,96,76),(61,124,89,77),(62,125,90,78),(63,126,91,79),(64,127,92,80)], [(1,62,43,90),(2,59,44,95),(3,64,45,92),(4,61,46,89),(5,58,47,94),(6,63,48,91),(7,60,41,96),(8,57,42,93),(9,98,114,24),(10,103,115,21),(11,100,116,18),(12,97,117,23),(13,102,118,20),(14,99,119,17),(15,104,120,22),(16,101,113,19),(25,75,106,122),(26,80,107,127),(27,77,108,124),(28,74,109,121),(29,79,110,126),(30,76,111,123),(31,73,112,128),(32,78,105,125),(33,67,85,50),(34,72,86,55),(35,69,87,52),(36,66,88,49),(37,71,81,54),(38,68,82,51),(39,65,83,56),(40,70,84,53)]])
Matrix representation of Q16⋊5Q8 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 15 | 10 | 0 | 0 | 0 | 0 |
| 8 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,8,0,0,2,0,0,0,0,0,0,9,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[4,0,0,0,0,0,2,13,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[15,8,0,0,0,0,10,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
Q16⋊5Q8 in GAP, Magma, Sage, TeX
Q_{16}\rtimes_5Q_8 % in TeX
G:=Group("Q16:5Q8"); // GroupNames label
G:=SmallGroup(128,2122);
// by ID
G=gap.SmallGroup(128,2122);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,723,352,346,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^4=1,b^2=a^4,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^5,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations
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