p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q16⋊4Q8, C42.68C23, C4.1022- 1+4, Q82.5C2, C8⋊Q8.2C2, C8.7(C2×Q8), C2.45(D4×Q8), C4⋊C4.390D4, Q8.12(C2×Q8), C8⋊4Q8.6C2, C8⋊3Q8.3C2, Q8.Q8.3C2, Q8⋊Q8.2C2, (C4×Q16).17C2, (C2×Q8).251D4, Q8⋊3Q8.7C2, C4.45(C22×Q8), C4⋊C8.146C22, C4⋊C4.276C23, (C4×C8).203C22, (C2×C4).579C24, (C2×C8).376C23, Q16⋊C4.2C2, C4.Q16.11C2, C4⋊Q8.208C22, C8⋊C4.72C22, C4.82(C8.C22), (C2×Q8).413C23, (C4×Q8).206C22, C4.Q8.119C22, C2.D8.140C22, C2.109(D4○SD16), Q8⋊C4.92C22, (C2×Q16).164C22, C22.839(C22×D4), C42.C2.77C22, (C2×C4).649(C2×D4), C2.91(C2×C8.C22), SmallGroup(128,2119)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q16⋊4Q8
G = < a,b,c,d | a8=c4=1, b2=a4, d2=c2, bab-1=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=c-1 >
Subgroups: 264 in 163 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C4, 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, C4×C8, C8⋊C4, Q8⋊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, C4⋊Q8, C2×Q16, C4×Q16, Q16⋊C4, C8⋊4Q8, Q8⋊Q8, Q8⋊Q8, C4.Q16, Q8.Q8, C8⋊3Q8, C8⋊Q8, Q8⋊3Q8, Q82, Q16⋊4Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C8.C22, C22×D4, C22×Q8, 2- 1+4, D4×Q8, C2×C8.C22, D4○SD16, Q16⋊4Q8
Character table of Q16⋊4Q8
| 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 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ26 | 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 |
| ρ27 | 4 | -4 | -4 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ28 | 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 | complex lifted from D4○SD16 |
| ρ29 | 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 | complex lifted from D4○SD16 |
►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 55 5 51)(2 54 6 50)(3 53 7 49)(4 52 8 56)(9 118 13 114)(10 117 14 113)(11 116 15 120)(12 115 16 119)(17 107 21 111)(18 106 22 110)(19 105 23 109)(20 112 24 108)(25 103 29 99)(26 102 30 98)(27 101 31 97)(28 100 32 104)(33 75 37 79)(34 74 38 78)(35 73 39 77)(36 80 40 76)(41 58 45 62)(42 57 46 61)(43 64 47 60)(44 63 48 59)(65 127 69 123)(66 126 70 122)(67 125 71 121)(68 124 72 128)(81 95 85 91)(82 94 86 90)(83 93 87 89)(84 92 88 96)
(1 29 42 107)(2 30 43 108)(3 31 44 109)(4 32 45 110)(5 25 46 111)(6 26 47 112)(7 27 48 105)(8 28 41 106)(9 79 124 95)(10 80 125 96)(11 73 126 89)(12 74 127 90)(13 75 128 91)(14 76 121 92)(15 77 122 93)(16 78 123 94)(17 51 103 61)(18 52 104 62)(19 53 97 63)(20 54 98 64)(21 55 99 57)(22 56 100 58)(23 49 101 59)(24 50 102 60)(33 72 85 118)(34 65 86 119)(35 66 87 120)(36 67 88 113)(37 68 81 114)(38 69 82 115)(39 70 83 116)(40 71 84 117)
(1 74 42 90)(2 79 43 95)(3 76 44 92)(4 73 45 89)(5 78 46 94)(6 75 47 91)(7 80 48 96)(8 77 41 93)(9 108 124 30)(10 105 125 27)(11 110 126 32)(12 107 127 29)(13 112 128 26)(14 109 121 31)(15 106 122 28)(16 111 123 25)(17 69 103 115)(18 66 104 120)(19 71 97 117)(20 68 98 114)(21 65 99 119)(22 70 100 116)(23 67 101 113)(24 72 102 118)(33 60 85 50)(34 57 86 55)(35 62 87 52)(36 59 88 49)(37 64 81 54)(38 61 82 51)(39 58 83 56)(40 63 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), (1,55,5,51)(2,54,6,50)(3,53,7,49)(4,52,8,56)(9,118,13,114)(10,117,14,113)(11,116,15,120)(12,115,16,119)(17,107,21,111)(18,106,22,110)(19,105,23,109)(20,112,24,108)(25,103,29,99)(26,102,30,98)(27,101,31,97)(28,100,32,104)(33,75,37,79)(34,74,38,78)(35,73,39,77)(36,80,40,76)(41,58,45,62)(42,57,46,61)(43,64,47,60)(44,63,48,59)(65,127,69,123)(66,126,70,122)(67,125,71,121)(68,124,72,128)(81,95,85,91)(82,94,86,90)(83,93,87,89)(84,92,88,96), (1,29,42,107)(2,30,43,108)(3,31,44,109)(4,32,45,110)(5,25,46,111)(6,26,47,112)(7,27,48,105)(8,28,41,106)(9,79,124,95)(10,80,125,96)(11,73,126,89)(12,74,127,90)(13,75,128,91)(14,76,121,92)(15,77,122,93)(16,78,123,94)(17,51,103,61)(18,52,104,62)(19,53,97,63)(20,54,98,64)(21,55,99,57)(22,56,100,58)(23,49,101,59)(24,50,102,60)(33,72,85,118)(34,65,86,119)(35,66,87,120)(36,67,88,113)(37,68,81,114)(38,69,82,115)(39,70,83,116)(40,71,84,117), (1,74,42,90)(2,79,43,95)(3,76,44,92)(4,73,45,89)(5,78,46,94)(6,75,47,91)(7,80,48,96)(8,77,41,93)(9,108,124,30)(10,105,125,27)(11,110,126,32)(12,107,127,29)(13,112,128,26)(14,109,121,31)(15,106,122,28)(16,111,123,25)(17,69,103,115)(18,66,104,120)(19,71,97,117)(20,68,98,114)(21,65,99,119)(22,70,100,116)(23,67,101,113)(24,72,102,118)(33,60,85,50)(34,57,86,55)(35,62,87,52)(36,59,88,49)(37,64,81,54)(38,61,82,51)(39,58,83,56)(40,63,84,53)>;
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), (1,55,5,51)(2,54,6,50)(3,53,7,49)(4,52,8,56)(9,118,13,114)(10,117,14,113)(11,116,15,120)(12,115,16,119)(17,107,21,111)(18,106,22,110)(19,105,23,109)(20,112,24,108)(25,103,29,99)(26,102,30,98)(27,101,31,97)(28,100,32,104)(33,75,37,79)(34,74,38,78)(35,73,39,77)(36,80,40,76)(41,58,45,62)(42,57,46,61)(43,64,47,60)(44,63,48,59)(65,127,69,123)(66,126,70,122)(67,125,71,121)(68,124,72,128)(81,95,85,91)(82,94,86,90)(83,93,87,89)(84,92,88,96), (1,29,42,107)(2,30,43,108)(3,31,44,109)(4,32,45,110)(5,25,46,111)(6,26,47,112)(7,27,48,105)(8,28,41,106)(9,79,124,95)(10,80,125,96)(11,73,126,89)(12,74,127,90)(13,75,128,91)(14,76,121,92)(15,77,122,93)(16,78,123,94)(17,51,103,61)(18,52,104,62)(19,53,97,63)(20,54,98,64)(21,55,99,57)(22,56,100,58)(23,49,101,59)(24,50,102,60)(33,72,85,118)(34,65,86,119)(35,66,87,120)(36,67,88,113)(37,68,81,114)(38,69,82,115)(39,70,83,116)(40,71,84,117), (1,74,42,90)(2,79,43,95)(3,76,44,92)(4,73,45,89)(5,78,46,94)(6,75,47,91)(7,80,48,96)(8,77,41,93)(9,108,124,30)(10,105,125,27)(11,110,126,32)(12,107,127,29)(13,112,128,26)(14,109,121,31)(15,106,122,28)(16,111,123,25)(17,69,103,115)(18,66,104,120)(19,71,97,117)(20,68,98,114)(21,65,99,119)(22,70,100,116)(23,67,101,113)(24,72,102,118)(33,60,85,50)(34,57,86,55)(35,62,87,52)(36,59,88,49)(37,64,81,54)(38,61,82,51)(39,58,83,56)(40,63,84,53) );
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)], [(1,55,5,51),(2,54,6,50),(3,53,7,49),(4,52,8,56),(9,118,13,114),(10,117,14,113),(11,116,15,120),(12,115,16,119),(17,107,21,111),(18,106,22,110),(19,105,23,109),(20,112,24,108),(25,103,29,99),(26,102,30,98),(27,101,31,97),(28,100,32,104),(33,75,37,79),(34,74,38,78),(35,73,39,77),(36,80,40,76),(41,58,45,62),(42,57,46,61),(43,64,47,60),(44,63,48,59),(65,127,69,123),(66,126,70,122),(67,125,71,121),(68,124,72,128),(81,95,85,91),(82,94,86,90),(83,93,87,89),(84,92,88,96)], [(1,29,42,107),(2,30,43,108),(3,31,44,109),(4,32,45,110),(5,25,46,111),(6,26,47,112),(7,27,48,105),(8,28,41,106),(9,79,124,95),(10,80,125,96),(11,73,126,89),(12,74,127,90),(13,75,128,91),(14,76,121,92),(15,77,122,93),(16,78,123,94),(17,51,103,61),(18,52,104,62),(19,53,97,63),(20,54,98,64),(21,55,99,57),(22,56,100,58),(23,49,101,59),(24,50,102,60),(33,72,85,118),(34,65,86,119),(35,66,87,120),(36,67,88,113),(37,68,81,114),(38,69,82,115),(39,70,83,116),(40,71,84,117)], [(1,74,42,90),(2,79,43,95),(3,76,44,92),(4,73,45,89),(5,78,46,94),(6,75,47,91),(7,80,48,96),(8,77,41,93),(9,108,124,30),(10,105,125,27),(11,110,126,32),(12,107,127,29),(13,112,128,26),(14,109,121,31),(15,106,122,28),(16,111,123,25),(17,69,103,115),(18,66,104,120),(19,71,97,117),(20,68,98,114),(21,65,99,119),(22,70,100,116),(23,67,101,113),(24,72,102,118),(33,60,85,50),(34,57,86,55),(35,62,87,52),(36,59,88,49),(37,64,81,54),(38,61,82,51),(39,58,83,56),(40,63,84,53)]])
Matrix representation of Q16⋊4Q8 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 10 |
| 0 | 0 | 5 | 12 | 10 | 7 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 12 | 10 | 12 | 10 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 11 | 3 | 4 |
| 0 | 0 | 4 | 15 | 10 | 10 |
| 0 | 0 | 4 | 10 | 16 | 16 |
| 0 | 0 | 5 | 14 | 12 | 15 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 | 1 | 15 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 4 | 8 | 12 |
| 0 | 0 | 16 | 12 | 3 | 6 |
| 0 | 0 | 16 | 9 | 10 | 13 |
| 0 | 0 | 3 | 6 | 13 | 9 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,5,12,12,0,0,5,12,5,10,0,0,0,10,0,12,0,0,10,7,0,10],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,5,4,4,5,0,0,11,15,10,14,0,0,3,10,16,12,0,0,4,10,16,15],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,16,16,0,0,0,16,0,1,0,0,1,1,0,0,0,0,0,15,0,1],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,3,16,16,3,0,0,4,12,9,6,0,0,8,3,10,13,0,0,12,6,13,9] >;
Q16⋊4Q8 in GAP, Magma, Sage, TeX
Q_{16}\rtimes_4Q_8 % in TeX
G:=Group("Q16:4Q8"); // GroupNames label
G:=SmallGroup(128,2119);
// by ID
G=gap.SmallGroup(128,2119);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,723,352,346,80,4037,1027,124]);
// 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,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations
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