direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8×Q16, C42.523C23, C4.972- 1+4, Q82.4C2, (C8×Q8).9C2, Q8.9(C2×Q8), C8.36(C2×Q8), C2.40(D4×Q8), C4⋊C4.282D4, (C4×Q16).9C2, C4.31(C2×Q16), C2.67(D4○D8), (C4×C8).96C22, (C2×Q8).272D4, C8⋊2Q8.16C2, C4.40(C22×Q8), C4⋊C4.271C23, C4⋊C8.305C22, (C2×C4).574C24, (C2×C8).212C23, C4.Q16.10C2, C4⋊Q8.203C22, C2.22(C22×Q16), C2.D8.73C22, (C2×Q8).409C23, (C4×Q8).202C22, (C2×Q16).172C22, C22.834(C22×D4), Q8⋊C4.164C22, (C2×C4).1104(C2×D4), SmallGroup(128,2114)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8×Q16
G = < a,b,c,d | a4=c8=1, b2=a2, d2=c4, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 280 in 168 conjugacy classes, 104 normal (14 characteristic)
C1, C2, C4, C4, C4, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, C2×Q8, C4×C8, Q8⋊C4, C4⋊C8, C2.D8, C4×Q8, C4×Q8, C4×Q8, C4⋊Q8, C4⋊Q8, C2×Q16, C4×Q16, C8×Q8, C4.Q16, C8⋊2Q8, Q82, Q8×Q16
Quotients: C1, C2, C22, D4, Q8, C23, Q16, C2×D4, C2×Q8, C24, C2×Q16, C22×D4, C22×Q8, 2- 1+4, D4×Q8, C22×Q16, D4○D8, Q8×Q16
►Regular action on 128 points
Generators in S128
(1 22 43 111)(2 23 44 112)(3 24 45 105)(4 17 46 106)(5 18 47 107)(6 19 48 108)(7 20 41 109)(8 21 42 110)(9 82 115 35)(10 83 116 36)(11 84 117 37)(12 85 118 38)(13 86 119 39)(14 87 120 40)(15 88 113 33)(16 81 114 34)(25 78 53 102)(26 79 54 103)(27 80 55 104)(28 73 56 97)(29 74 49 98)(30 75 50 99)(31 76 51 100)(32 77 52 101)(57 122 89 68)(58 123 90 69)(59 124 91 70)(60 125 92 71)(61 126 93 72)(62 127 94 65)(63 128 95 66)(64 121 96 67)
(1 58 43 90)(2 59 44 91)(3 60 45 92)(4 61 46 93)(5 62 47 94)(6 63 48 95)(7 64 41 96)(8 57 42 89)(9 103 115 79)(10 104 116 80)(11 97 117 73)(12 98 118 74)(13 99 119 75)(14 100 120 76)(15 101 113 77)(16 102 114 78)(17 72 106 126)(18 65 107 127)(19 66 108 128)(20 67 109 121)(21 68 110 122)(22 69 111 123)(23 70 112 124)(24 71 105 125)(25 81 53 34)(26 82 54 35)(27 83 55 36)(28 84 56 37)(29 85 49 38)(30 86 50 39)(31 87 51 40)(32 88 52 33)
(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 54 5 50)(2 53 6 49)(3 52 7 56)(4 51 8 55)(9 127 13 123)(10 126 14 122)(11 125 15 121)(12 124 16 128)(17 100 21 104)(18 99 22 103)(19 98 23 102)(20 97 24 101)(25 48 29 44)(26 47 30 43)(27 46 31 42)(28 45 32 41)(33 64 37 60)(34 63 38 59)(35 62 39 58)(36 61 40 57)(65 119 69 115)(66 118 70 114)(67 117 71 113)(68 116 72 120)(73 105 77 109)(74 112 78 108)(75 111 79 107)(76 110 80 106)(81 95 85 91)(82 94 86 90)(83 93 87 89)(84 92 88 96)
G:=sub<Sym(128)| (1,22,43,111)(2,23,44,112)(3,24,45,105)(4,17,46,106)(5,18,47,107)(6,19,48,108)(7,20,41,109)(8,21,42,110)(9,82,115,35)(10,83,116,36)(11,84,117,37)(12,85,118,38)(13,86,119,39)(14,87,120,40)(15,88,113,33)(16,81,114,34)(25,78,53,102)(26,79,54,103)(27,80,55,104)(28,73,56,97)(29,74,49,98)(30,75,50,99)(31,76,51,100)(32,77,52,101)(57,122,89,68)(58,123,90,69)(59,124,91,70)(60,125,92,71)(61,126,93,72)(62,127,94,65)(63,128,95,66)(64,121,96,67), (1,58,43,90)(2,59,44,91)(3,60,45,92)(4,61,46,93)(5,62,47,94)(6,63,48,95)(7,64,41,96)(8,57,42,89)(9,103,115,79)(10,104,116,80)(11,97,117,73)(12,98,118,74)(13,99,119,75)(14,100,120,76)(15,101,113,77)(16,102,114,78)(17,72,106,126)(18,65,107,127)(19,66,108,128)(20,67,109,121)(21,68,110,122)(22,69,111,123)(23,70,112,124)(24,71,105,125)(25,81,53,34)(26,82,54,35)(27,83,55,36)(28,84,56,37)(29,85,49,38)(30,86,50,39)(31,87,51,40)(32,88,52,33), (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), (1,54,5,50)(2,53,6,49)(3,52,7,56)(4,51,8,55)(9,127,13,123)(10,126,14,122)(11,125,15,121)(12,124,16,128)(17,100,21,104)(18,99,22,103)(19,98,23,102)(20,97,24,101)(25,48,29,44)(26,47,30,43)(27,46,31,42)(28,45,32,41)(33,64,37,60)(34,63,38,59)(35,62,39,58)(36,61,40,57)(65,119,69,115)(66,118,70,114)(67,117,71,113)(68,116,72,120)(73,105,77,109)(74,112,78,108)(75,111,79,107)(76,110,80,106)(81,95,85,91)(82,94,86,90)(83,93,87,89)(84,92,88,96)>;
G:=Group( (1,22,43,111)(2,23,44,112)(3,24,45,105)(4,17,46,106)(5,18,47,107)(6,19,48,108)(7,20,41,109)(8,21,42,110)(9,82,115,35)(10,83,116,36)(11,84,117,37)(12,85,118,38)(13,86,119,39)(14,87,120,40)(15,88,113,33)(16,81,114,34)(25,78,53,102)(26,79,54,103)(27,80,55,104)(28,73,56,97)(29,74,49,98)(30,75,50,99)(31,76,51,100)(32,77,52,101)(57,122,89,68)(58,123,90,69)(59,124,91,70)(60,125,92,71)(61,126,93,72)(62,127,94,65)(63,128,95,66)(64,121,96,67), (1,58,43,90)(2,59,44,91)(3,60,45,92)(4,61,46,93)(5,62,47,94)(6,63,48,95)(7,64,41,96)(8,57,42,89)(9,103,115,79)(10,104,116,80)(11,97,117,73)(12,98,118,74)(13,99,119,75)(14,100,120,76)(15,101,113,77)(16,102,114,78)(17,72,106,126)(18,65,107,127)(19,66,108,128)(20,67,109,121)(21,68,110,122)(22,69,111,123)(23,70,112,124)(24,71,105,125)(25,81,53,34)(26,82,54,35)(27,83,55,36)(28,84,56,37)(29,85,49,38)(30,86,50,39)(31,87,51,40)(32,88,52,33), (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), (1,54,5,50)(2,53,6,49)(3,52,7,56)(4,51,8,55)(9,127,13,123)(10,126,14,122)(11,125,15,121)(12,124,16,128)(17,100,21,104)(18,99,22,103)(19,98,23,102)(20,97,24,101)(25,48,29,44)(26,47,30,43)(27,46,31,42)(28,45,32,41)(33,64,37,60)(34,63,38,59)(35,62,39,58)(36,61,40,57)(65,119,69,115)(66,118,70,114)(67,117,71,113)(68,116,72,120)(73,105,77,109)(74,112,78,108)(75,111,79,107)(76,110,80,106)(81,95,85,91)(82,94,86,90)(83,93,87,89)(84,92,88,96) );
G=PermutationGroup([[(1,22,43,111),(2,23,44,112),(3,24,45,105),(4,17,46,106),(5,18,47,107),(6,19,48,108),(7,20,41,109),(8,21,42,110),(9,82,115,35),(10,83,116,36),(11,84,117,37),(12,85,118,38),(13,86,119,39),(14,87,120,40),(15,88,113,33),(16,81,114,34),(25,78,53,102),(26,79,54,103),(27,80,55,104),(28,73,56,97),(29,74,49,98),(30,75,50,99),(31,76,51,100),(32,77,52,101),(57,122,89,68),(58,123,90,69),(59,124,91,70),(60,125,92,71),(61,126,93,72),(62,127,94,65),(63,128,95,66),(64,121,96,67)], [(1,58,43,90),(2,59,44,91),(3,60,45,92),(4,61,46,93),(5,62,47,94),(6,63,48,95),(7,64,41,96),(8,57,42,89),(9,103,115,79),(10,104,116,80),(11,97,117,73),(12,98,118,74),(13,99,119,75),(14,100,120,76),(15,101,113,77),(16,102,114,78),(17,72,106,126),(18,65,107,127),(19,66,108,128),(20,67,109,121),(21,68,110,122),(22,69,111,123),(23,70,112,124),(24,71,105,125),(25,81,53,34),(26,82,54,35),(27,83,55,36),(28,84,56,37),(29,85,49,38),(30,86,50,39),(31,87,51,40),(32,88,52,33)], [(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)], [(1,54,5,50),(2,53,6,49),(3,52,7,56),(4,51,8,55),(9,127,13,123),(10,126,14,122),(11,125,15,121),(12,124,16,128),(17,100,21,104),(18,99,22,103),(19,98,23,102),(20,97,24,101),(25,48,29,44),(26,47,30,43),(27,46,31,42),(28,45,32,41),(33,64,37,60),(34,63,38,59),(35,62,39,58),(36,61,40,57),(65,119,69,115),(66,118,70,114),(67,117,71,113),(68,116,72,120),(73,105,77,109),(74,112,78,108),(75,111,79,107),(76,110,80,106),(81,95,85,91),(82,94,86,90),(83,93,87,89),(84,92,88,96)]])
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 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | - | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | Q16 | 2- 1+4 | D4○D8 |
| kernel | Q8×Q16 | C4×Q16 | C8×Q8 | C4.Q16 | C8⋊2Q8 | Q82 | C4⋊C4 | Q16 | C2×Q8 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 1 | 6 | 3 | 2 | 3 | 4 | 1 | 8 | 1 | 2 |
Matrix representation of Q8×Q16 ►in GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 12 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 3 | 11 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 7 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,1,0,0,0,0,1],[5,12,0,0,12,12,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,0,3,0,0,11,11],[16,0,0,0,0,16,0,0,0,0,0,12,0,0,7,0] >;
Q8×Q16 in GAP, Magma, Sage, TeX
Q_8\times Q_{16} % in TeX
G:=Group("Q8xQ16"); // GroupNames label
G:=SmallGroup(128,2114);
// by ID
G=gap.SmallGroup(128,2114);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,352,346,80,4037,1027,124]);
// 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,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations