p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊5Q16, C42.504C23, C4.252- 1+4, Q82.3C2, (C8×Q8).8C2, C4⋊C4.277D4, (C4×Q16).8C2, C4.30(C2×Q16), Q8○3(Q8⋊C4), (C4×C8).93C22, (C2×Q8).269D4, C4.Q16.9C2, Q8⋊3Q8.4C2, C4⋊C4.431C23, C4⋊C8.303C22, (C2×C8).206C23, (C2×C4).555C24, Q8.33(C4○D4), C4⋊2Q16.10C2, C4⋊Q8.184C22, C4.SD16.8C2, C2.21(C22×Q16), C2.63(Q8⋊5D4), C2.98(D4○SD16), (C4×Q8).309C22, (C2×Q8).253C23, C2.D8.201C22, (C2×Q16).141C22, Q8⋊C4.18C22, C22.815(C22×D4), C4.256(C2×C4○D4), (C2×C4).1101(C2×D4), SmallGroup(128,2095)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8⋊5Q16
G = < a,b,c,d | a4=c8=1, b2=a2, d2=c4, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=c-1 >
Subgroups: 272 in 168 conjugacy classes, 96 normal (18 characteristic)
C1, C2, C4, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C42, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, C2×Q8, C4×C8, Q8⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C4×Q8, C4×Q8, C4×Q8, C42.C2, C4⋊Q8, C4⋊Q8, C2×Q16, C4×Q16, C8×Q8, C4⋊2Q16, C4.Q16, C4.SD16, Q8⋊3Q8, Q82, Q8⋊5Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C4○D4, C24, C2×Q16, C22×D4, C2×C4○D4, 2- 1+4, Q8⋊5D4, C22×Q16, D4○SD16, Q8⋊5Q16
►Regular action on 128 points
Generators in S128
(1 39 107 89)(2 40 108 90)(3 33 109 91)(4 34 110 92)(5 35 111 93)(6 36 112 94)(7 37 105 95)(8 38 106 96)(9 115 47 56)(10 116 48 49)(11 117 41 50)(12 118 42 51)(13 119 43 52)(14 120 44 53)(15 113 45 54)(16 114 46 55)(17 125 84 77)(18 126 85 78)(19 127 86 79)(20 128 87 80)(21 121 88 73)(22 122 81 74)(23 123 82 75)(24 124 83 76)(25 63 100 71)(26 64 101 72)(27 57 102 65)(28 58 103 66)(29 59 104 67)(30 60 97 68)(31 61 98 69)(32 62 99 70)
(1 127 107 79)(2 80 108 128)(3 121 109 73)(4 74 110 122)(5 123 111 75)(6 76 112 124)(7 125 105 77)(8 78 106 126)(9 66 47 58)(10 59 48 67)(11 68 41 60)(12 61 42 69)(13 70 43 62)(14 63 44 71)(15 72 45 64)(16 57 46 65)(17 95 84 37)(18 38 85 96)(19 89 86 39)(20 40 87 90)(21 91 88 33)(22 34 81 92)(23 93 82 35)(24 36 83 94)(25 53 100 120)(26 113 101 54)(27 55 102 114)(28 115 103 56)(29 49 104 116)(30 117 97 50)(31 51 98 118)(32 119 99 52)
(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 28 5 32)(2 27 6 31)(3 26 7 30)(4 25 8 29)(9 23 13 19)(10 22 14 18)(11 21 15 17)(12 20 16 24)(33 64 37 60)(34 63 38 59)(35 62 39 58)(36 61 40 57)(41 88 45 84)(42 87 46 83)(43 86 47 82)(44 85 48 81)(49 74 53 78)(50 73 54 77)(51 80 55 76)(52 79 56 75)(65 94 69 90)(66 93 70 89)(67 92 71 96)(68 91 72 95)(97 109 101 105)(98 108 102 112)(99 107 103 111)(100 106 104 110)(113 125 117 121)(114 124 118 128)(115 123 119 127)(116 122 120 126)
G:=sub<Sym(128)| (1,39,107,89)(2,40,108,90)(3,33,109,91)(4,34,110,92)(5,35,111,93)(6,36,112,94)(7,37,105,95)(8,38,106,96)(9,115,47,56)(10,116,48,49)(11,117,41,50)(12,118,42,51)(13,119,43,52)(14,120,44,53)(15,113,45,54)(16,114,46,55)(17,125,84,77)(18,126,85,78)(19,127,86,79)(20,128,87,80)(21,121,88,73)(22,122,81,74)(23,123,82,75)(24,124,83,76)(25,63,100,71)(26,64,101,72)(27,57,102,65)(28,58,103,66)(29,59,104,67)(30,60,97,68)(31,61,98,69)(32,62,99,70), (1,127,107,79)(2,80,108,128)(3,121,109,73)(4,74,110,122)(5,123,111,75)(6,76,112,124)(7,125,105,77)(8,78,106,126)(9,66,47,58)(10,59,48,67)(11,68,41,60)(12,61,42,69)(13,70,43,62)(14,63,44,71)(15,72,45,64)(16,57,46,65)(17,95,84,37)(18,38,85,96)(19,89,86,39)(20,40,87,90)(21,91,88,33)(22,34,81,92)(23,93,82,35)(24,36,83,94)(25,53,100,120)(26,113,101,54)(27,55,102,114)(28,115,103,56)(29,49,104,116)(30,117,97,50)(31,51,98,118)(32,119,99,52), (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), (1,28,5,32)(2,27,6,31)(3,26,7,30)(4,25,8,29)(9,23,13,19)(10,22,14,18)(11,21,15,17)(12,20,16,24)(33,64,37,60)(34,63,38,59)(35,62,39,58)(36,61,40,57)(41,88,45,84)(42,87,46,83)(43,86,47,82)(44,85,48,81)(49,74,53,78)(50,73,54,77)(51,80,55,76)(52,79,56,75)(65,94,69,90)(66,93,70,89)(67,92,71,96)(68,91,72,95)(97,109,101,105)(98,108,102,112)(99,107,103,111)(100,106,104,110)(113,125,117,121)(114,124,118,128)(115,123,119,127)(116,122,120,126)>;
G:=Group( (1,39,107,89)(2,40,108,90)(3,33,109,91)(4,34,110,92)(5,35,111,93)(6,36,112,94)(7,37,105,95)(8,38,106,96)(9,115,47,56)(10,116,48,49)(11,117,41,50)(12,118,42,51)(13,119,43,52)(14,120,44,53)(15,113,45,54)(16,114,46,55)(17,125,84,77)(18,126,85,78)(19,127,86,79)(20,128,87,80)(21,121,88,73)(22,122,81,74)(23,123,82,75)(24,124,83,76)(25,63,100,71)(26,64,101,72)(27,57,102,65)(28,58,103,66)(29,59,104,67)(30,60,97,68)(31,61,98,69)(32,62,99,70), (1,127,107,79)(2,80,108,128)(3,121,109,73)(4,74,110,122)(5,123,111,75)(6,76,112,124)(7,125,105,77)(8,78,106,126)(9,66,47,58)(10,59,48,67)(11,68,41,60)(12,61,42,69)(13,70,43,62)(14,63,44,71)(15,72,45,64)(16,57,46,65)(17,95,84,37)(18,38,85,96)(19,89,86,39)(20,40,87,90)(21,91,88,33)(22,34,81,92)(23,93,82,35)(24,36,83,94)(25,53,100,120)(26,113,101,54)(27,55,102,114)(28,115,103,56)(29,49,104,116)(30,117,97,50)(31,51,98,118)(32,119,99,52), (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), (1,28,5,32)(2,27,6,31)(3,26,7,30)(4,25,8,29)(9,23,13,19)(10,22,14,18)(11,21,15,17)(12,20,16,24)(33,64,37,60)(34,63,38,59)(35,62,39,58)(36,61,40,57)(41,88,45,84)(42,87,46,83)(43,86,47,82)(44,85,48,81)(49,74,53,78)(50,73,54,77)(51,80,55,76)(52,79,56,75)(65,94,69,90)(66,93,70,89)(67,92,71,96)(68,91,72,95)(97,109,101,105)(98,108,102,112)(99,107,103,111)(100,106,104,110)(113,125,117,121)(114,124,118,128)(115,123,119,127)(116,122,120,126) );
G=PermutationGroup([[(1,39,107,89),(2,40,108,90),(3,33,109,91),(4,34,110,92),(5,35,111,93),(6,36,112,94),(7,37,105,95),(8,38,106,96),(9,115,47,56),(10,116,48,49),(11,117,41,50),(12,118,42,51),(13,119,43,52),(14,120,44,53),(15,113,45,54),(16,114,46,55),(17,125,84,77),(18,126,85,78),(19,127,86,79),(20,128,87,80),(21,121,88,73),(22,122,81,74),(23,123,82,75),(24,124,83,76),(25,63,100,71),(26,64,101,72),(27,57,102,65),(28,58,103,66),(29,59,104,67),(30,60,97,68),(31,61,98,69),(32,62,99,70)], [(1,127,107,79),(2,80,108,128),(3,121,109,73),(4,74,110,122),(5,123,111,75),(6,76,112,124),(7,125,105,77),(8,78,106,126),(9,66,47,58),(10,59,48,67),(11,68,41,60),(12,61,42,69),(13,70,43,62),(14,63,44,71),(15,72,45,64),(16,57,46,65),(17,95,84,37),(18,38,85,96),(19,89,86,39),(20,40,87,90),(21,91,88,33),(22,34,81,92),(23,93,82,35),(24,36,83,94),(25,53,100,120),(26,113,101,54),(27,55,102,114),(28,115,103,56),(29,49,104,116),(30,117,97,50),(31,51,98,118),(32,119,99,52)], [(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)], [(1,28,5,32),(2,27,6,31),(3,26,7,30),(4,25,8,29),(9,23,13,19),(10,22,14,18),(11,21,15,17),(12,20,16,24),(33,64,37,60),(34,63,38,59),(35,62,39,58),(36,61,40,57),(41,88,45,84),(42,87,46,83),(43,86,47,82),(44,85,48,81),(49,74,53,78),(50,73,54,77),(51,80,55,76),(52,79,56,75),(65,94,69,90),(66,93,70,89),(67,92,71,96),(68,91,72,95),(97,109,101,105),(98,108,102,112),(99,107,103,111),(100,106,104,110),(113,125,117,121),(114,124,118,128),(115,123,119,127),(116,122,120,126)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4H | 4I | ··· | 4O | 4P | ··· | 4U | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q16 | C4○D4 | 2- 1+4 | D4○SD16 |
| kernel | Q8⋊5Q16 | C4×Q16 | C8×Q8 | C4⋊2Q16 | C4.Q16 | C4.SD16 | Q8⋊3Q8 | Q82 | C4⋊C4 | C2×Q8 | Q8 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 1 | 3 | 3 | 3 | 1 | 1 | 3 | 1 | 8 | 4 | 1 | 2 |
Matrix representation of Q8⋊5Q16 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 7 | 1 |
| 0 | 0 | 1 | 10 |
| 8 | 0 | 0 | 0 |
| 5 | 15 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
| 1 | 15 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,16,0,0,0,0,7,1,0,0,1,10],[8,5,0,0,0,15,0,0,0,0,0,4,0,0,13,0],[1,1,0,0,15,16,0,0,0,0,16,0,0,0,0,16] >;
Q8⋊5Q16 in GAP, Magma, Sage, TeX
Q_8\rtimes_5Q_{16} % in TeX
G:=Group("Q8:5Q16"); // GroupNames label
G:=SmallGroup(128,2095);
// by ID
G=gap.SmallGroup(128,2095);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,352,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=1,b^2=a^2,d^2=c^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations