direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: Q8×C16, C16○3(C4⋊C4), C16○2(C4⋊C8), C4⋊C8.28C4, C4⋊C4.14C8, C2.2(C8×Q8), (C4×C16).5C2, C4.4(C2×C16), C16○2(C4⋊C16), C4⋊C16.14C2, C8.43(C2×Q8), C4.47(C4×Q8), (C4×Q8).37C4, (C2×Q8).11C8, (C8×Q8).17C2, C4.60(C8○D4), C2.3(D4○C16), C2.5(C22×C16), C8.104(C4○D4), (C4×C8).377C22, C42.275(C2×C4), (C2×C8).635C23, (C2×C16).112C22, C22.30(C22×C8), (C2×C16)○(C4×Q8), (C2×C16)○(C8×Q8), (C2×C4).42(C2×C8), (C2×C8).164(C2×C4), (C2×C4).620(C22×C4), SmallGroup(128,914)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8×C16
G = < a,b,c | a16=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 76 in 67 conjugacy classes, 58 normal (16 characteristic)
C1, C2 [×3], C4 [×2], C4 [×6], C4 [×3], C22, C8 [×2], C8 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C16 [×2], C16 [×3], C42 [×3], C4⋊C4 [×3], C2×C8, C2×C8 [×3], C2×Q8, C4×C8 [×3], C4⋊C8 [×3], C2×C16, C2×C16 [×3], C4×Q8, C4×C16 [×3], C4⋊C16 [×3], C8×Q8, Q8×C16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], Q8 [×2], C23, C16 [×4], C2×C8 [×6], C22×C4, C2×Q8, C4○D4, C2×C16 [×6], C4×Q8, C22×C8, C8○D4, C8×Q8, C22×C16, D4○C16, Q8×C16
►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 64 97 119)(2 49 98 120)(3 50 99 121)(4 51 100 122)(5 52 101 123)(6 53 102 124)(7 54 103 125)(8 55 104 126)(9 56 105 127)(10 57 106 128)(11 58 107 113)(12 59 108 114)(13 60 109 115)(14 61 110 116)(15 62 111 117)(16 63 112 118)(17 69 92 38)(18 70 93 39)(19 71 94 40)(20 72 95 41)(21 73 96 42)(22 74 81 43)(23 75 82 44)(24 76 83 45)(25 77 84 46)(26 78 85 47)(27 79 86 48)(28 80 87 33)(29 65 88 34)(30 66 89 35)(31 67 90 36)(32 68 91 37)
(1 76 97 45)(2 77 98 46)(3 78 99 47)(4 79 100 48)(5 80 101 33)(6 65 102 34)(7 66 103 35)(8 67 104 36)(9 68 105 37)(10 69 106 38)(11 70 107 39)(12 71 108 40)(13 72 109 41)(14 73 110 42)(15 74 111 43)(16 75 112 44)(17 128 92 57)(18 113 93 58)(19 114 94 59)(20 115 95 60)(21 116 96 61)(22 117 81 62)(23 118 82 63)(24 119 83 64)(25 120 84 49)(26 121 85 50)(27 122 86 51)(28 123 87 52)(29 124 88 53)(30 125 89 54)(31 126 90 55)(32 127 91 56)
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), (1,64,97,119)(2,49,98,120)(3,50,99,121)(4,51,100,122)(5,52,101,123)(6,53,102,124)(7,54,103,125)(8,55,104,126)(9,56,105,127)(10,57,106,128)(11,58,107,113)(12,59,108,114)(13,60,109,115)(14,61,110,116)(15,62,111,117)(16,63,112,118)(17,69,92,38)(18,70,93,39)(19,71,94,40)(20,72,95,41)(21,73,96,42)(22,74,81,43)(23,75,82,44)(24,76,83,45)(25,77,84,46)(26,78,85,47)(27,79,86,48)(28,80,87,33)(29,65,88,34)(30,66,89,35)(31,67,90,36)(32,68,91,37), (1,76,97,45)(2,77,98,46)(3,78,99,47)(4,79,100,48)(5,80,101,33)(6,65,102,34)(7,66,103,35)(8,67,104,36)(9,68,105,37)(10,69,106,38)(11,70,107,39)(12,71,108,40)(13,72,109,41)(14,73,110,42)(15,74,111,43)(16,75,112,44)(17,128,92,57)(18,113,93,58)(19,114,94,59)(20,115,95,60)(21,116,96,61)(22,117,81,62)(23,118,82,63)(24,119,83,64)(25,120,84,49)(26,121,85,50)(27,122,86,51)(28,123,87,52)(29,124,88,53)(30,125,89,54)(31,126,90,55)(32,127,91,56)>;
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), (1,64,97,119)(2,49,98,120)(3,50,99,121)(4,51,100,122)(5,52,101,123)(6,53,102,124)(7,54,103,125)(8,55,104,126)(9,56,105,127)(10,57,106,128)(11,58,107,113)(12,59,108,114)(13,60,109,115)(14,61,110,116)(15,62,111,117)(16,63,112,118)(17,69,92,38)(18,70,93,39)(19,71,94,40)(20,72,95,41)(21,73,96,42)(22,74,81,43)(23,75,82,44)(24,76,83,45)(25,77,84,46)(26,78,85,47)(27,79,86,48)(28,80,87,33)(29,65,88,34)(30,66,89,35)(31,67,90,36)(32,68,91,37), (1,76,97,45)(2,77,98,46)(3,78,99,47)(4,79,100,48)(5,80,101,33)(6,65,102,34)(7,66,103,35)(8,67,104,36)(9,68,105,37)(10,69,106,38)(11,70,107,39)(12,71,108,40)(13,72,109,41)(14,73,110,42)(15,74,111,43)(16,75,112,44)(17,128,92,57)(18,113,93,58)(19,114,94,59)(20,115,95,60)(21,116,96,61)(22,117,81,62)(23,118,82,63)(24,119,83,64)(25,120,84,49)(26,121,85,50)(27,122,86,51)(28,123,87,52)(29,124,88,53)(30,125,89,54)(31,126,90,55)(32,127,91,56) );
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)], [(1,64,97,119),(2,49,98,120),(3,50,99,121),(4,51,100,122),(5,52,101,123),(6,53,102,124),(7,54,103,125),(8,55,104,126),(9,56,105,127),(10,57,106,128),(11,58,107,113),(12,59,108,114),(13,60,109,115),(14,61,110,116),(15,62,111,117),(16,63,112,118),(17,69,92,38),(18,70,93,39),(19,71,94,40),(20,72,95,41),(21,73,96,42),(22,74,81,43),(23,75,82,44),(24,76,83,45),(25,77,84,46),(26,78,85,47),(27,79,86,48),(28,80,87,33),(29,65,88,34),(30,66,89,35),(31,67,90,36),(32,68,91,37)], [(1,76,97,45),(2,77,98,46),(3,78,99,47),(4,79,100,48),(5,80,101,33),(6,65,102,34),(7,66,103,35),(8,67,104,36),(9,68,105,37),(10,69,106,38),(11,70,107,39),(12,71,108,40),(13,72,109,41),(14,73,110,42),(15,74,111,43),(16,75,112,44),(17,128,92,57),(18,113,93,58),(19,114,94,59),(20,115,95,60),(21,116,96,61),(22,117,81,62),(23,118,82,63),(24,119,83,64),(25,120,84,49),(26,121,85,50),(27,122,86,51),(28,123,87,52),(29,124,88,53),(30,125,89,54),(31,126,90,55),(32,127,91,56)])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 8A | ··· | 8H | 8I | ··· | 8T | 16A | ··· | 16P | 16Q | ··· | 16AN |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | Q8 | C4○D4 | C8○D4 | D4○C16 |
| kernel | Q8×C16 | C4×C16 | C4⋊C16 | C8×Q8 | C4⋊C8 | C4×Q8 | C4⋊C4 | C2×Q8 | Q8 | C16 | C8 | C4 | C2 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 12 | 4 | 32 | 2 | 2 | 4 | 8 |
Matrix representation of Q8×C16 ►in GL3(𝔽17) generated by
| 10 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 16 | 0 |
| 1 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 4 |
G:=sub<GL(3,GF(17))| [10,0,0,0,2,0,0,0,2],[1,0,0,0,0,16,0,1,0],[1,0,0,0,13,0,0,0,4] >;
Q8×C16 in GAP, Magma, Sage, TeX
Q_8\times C_{16} % in TeX
G:=Group("Q8xC16"); // GroupNames label
G:=SmallGroup(128,914);
// by ID
G=gap.SmallGroup(128,914);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,142,102,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations