direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8×Q16, C82.3C2, C42.634C23, C2.6(C8×D4), C8○2(Q8⋊C8), C8.10(C2×C8), C8○2(C8⋊1C8), C8○2(C2.D8), (C8×Q8).2C2, Q8.1(C2×C8), C2.1(C4×Q16), C4.9(C8○D4), C2.3(C8○D8), (C2×C8).219D4, Q8⋊C8.11C2, C4.9(C22×C8), C4.57(C2×Q16), C8⋊1C8.15C2, C2.D8.23C4, C8○2(Q8⋊C4), (C4×Q16).21C2, (C2×Q16).15C4, C22.79(C4×D4), C4.127(C4○D8), C4⋊C8.272C22, (C4×C8).368C22, Q8⋊C4.14C4, (C4×Q8).256C22, (C2×C8)○(C4×Q16), C4⋊C4.128(C2×C4), (C2×C8).156(C2×C4), (C2×C4).1470(C2×D4), (C2×Q8).132(C2×C4), (C2×C4).495(C4○D4), (C2×C4).326(C22×C4), SmallGroup(128,309)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8×Q16
G = < a,b,c | a8=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 120 in 82 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C4, C4, C22, C8, C8, C2×C4, C2×C4, Q8, Q8, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C4×C8, C4×C8, Q8⋊C4, C4⋊C8, C4⋊C8, C2.D8, C4×Q8, C2×Q16, C82, Q8⋊C8, C8⋊1C8, C4×Q16, C8×Q8, C8×Q16
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, Q16, C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C2×Q16, C4○D8, C8×D4, C4×Q16, C8○D8, C8×Q16
►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)
(1 111 39 124 9 20 31 116)(2 112 40 125 10 21 32 117)(3 105 33 126 11 22 25 118)(4 106 34 127 12 23 26 119)(5 107 35 128 13 24 27 120)(6 108 36 121 14 17 28 113)(7 109 37 122 15 18 29 114)(8 110 38 123 16 19 30 115)(41 83 54 99 70 91 62 78)(42 84 55 100 71 92 63 79)(43 85 56 101 72 93 64 80)(44 86 49 102 65 94 57 73)(45 87 50 103 66 95 58 74)(46 88 51 104 67 96 59 75)(47 81 52 97 68 89 60 76)(48 82 53 98 69 90 61 77)
(1 51 9 59)(2 52 10 60)(3 53 11 61)(4 54 12 62)(5 55 13 63)(6 56 14 64)(7 49 15 57)(8 50 16 58)(17 93 108 85)(18 94 109 86)(19 95 110 87)(20 96 111 88)(21 89 112 81)(22 90 105 82)(23 91 106 83)(24 92 107 84)(25 69 33 48)(26 70 34 41)(27 71 35 42)(28 72 36 43)(29 65 37 44)(30 66 38 45)(31 67 39 46)(32 68 40 47)(73 114 102 122)(74 115 103 123)(75 116 104 124)(76 117 97 125)(77 118 98 126)(78 119 99 127)(79 120 100 128)(80 113 101 121)
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), (1,111,39,124,9,20,31,116)(2,112,40,125,10,21,32,117)(3,105,33,126,11,22,25,118)(4,106,34,127,12,23,26,119)(5,107,35,128,13,24,27,120)(6,108,36,121,14,17,28,113)(7,109,37,122,15,18,29,114)(8,110,38,123,16,19,30,115)(41,83,54,99,70,91,62,78)(42,84,55,100,71,92,63,79)(43,85,56,101,72,93,64,80)(44,86,49,102,65,94,57,73)(45,87,50,103,66,95,58,74)(46,88,51,104,67,96,59,75)(47,81,52,97,68,89,60,76)(48,82,53,98,69,90,61,77), (1,51,9,59)(2,52,10,60)(3,53,11,61)(4,54,12,62)(5,55,13,63)(6,56,14,64)(7,49,15,57)(8,50,16,58)(17,93,108,85)(18,94,109,86)(19,95,110,87)(20,96,111,88)(21,89,112,81)(22,90,105,82)(23,91,106,83)(24,92,107,84)(25,69,33,48)(26,70,34,41)(27,71,35,42)(28,72,36,43)(29,65,37,44)(30,66,38,45)(31,67,39,46)(32,68,40,47)(73,114,102,122)(74,115,103,123)(75,116,104,124)(76,117,97,125)(77,118,98,126)(78,119,99,127)(79,120,100,128)(80,113,101,121)>;
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), (1,111,39,124,9,20,31,116)(2,112,40,125,10,21,32,117)(3,105,33,126,11,22,25,118)(4,106,34,127,12,23,26,119)(5,107,35,128,13,24,27,120)(6,108,36,121,14,17,28,113)(7,109,37,122,15,18,29,114)(8,110,38,123,16,19,30,115)(41,83,54,99,70,91,62,78)(42,84,55,100,71,92,63,79)(43,85,56,101,72,93,64,80)(44,86,49,102,65,94,57,73)(45,87,50,103,66,95,58,74)(46,88,51,104,67,96,59,75)(47,81,52,97,68,89,60,76)(48,82,53,98,69,90,61,77), (1,51,9,59)(2,52,10,60)(3,53,11,61)(4,54,12,62)(5,55,13,63)(6,56,14,64)(7,49,15,57)(8,50,16,58)(17,93,108,85)(18,94,109,86)(19,95,110,87)(20,96,111,88)(21,89,112,81)(22,90,105,82)(23,91,106,83)(24,92,107,84)(25,69,33,48)(26,70,34,41)(27,71,35,42)(28,72,36,43)(29,65,37,44)(30,66,38,45)(31,67,39,46)(32,68,40,47)(73,114,102,122)(74,115,103,123)(75,116,104,124)(76,117,97,125)(77,118,98,126)(78,119,99,127)(79,120,100,128)(80,113,101,121) );
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)], [(1,111,39,124,9,20,31,116),(2,112,40,125,10,21,32,117),(3,105,33,126,11,22,25,118),(4,106,34,127,12,23,26,119),(5,107,35,128,13,24,27,120),(6,108,36,121,14,17,28,113),(7,109,37,122,15,18,29,114),(8,110,38,123,16,19,30,115),(41,83,54,99,70,91,62,78),(42,84,55,100,71,92,63,79),(43,85,56,101,72,93,64,80),(44,86,49,102,65,94,57,73),(45,87,50,103,66,95,58,74),(46,88,51,104,67,96,59,75),(47,81,52,97,68,89,60,76),(48,82,53,98,69,90,61,77)], [(1,51,9,59),(2,52,10,60),(3,53,11,61),(4,54,12,62),(5,55,13,63),(6,56,14,64),(7,49,15,57),(8,50,16,58),(17,93,108,85),(18,94,109,86),(19,95,110,87),(20,96,111,88),(21,89,112,81),(22,90,105,82),(23,91,106,83),(24,92,107,84),(25,69,33,48),(26,70,34,41),(27,71,35,42),(28,72,36,43),(29,65,37,44),(30,66,38,45),(31,67,39,46),(32,68,40,47),(73,114,102,122),(74,115,103,123),(75,116,104,124),(76,117,97,125),(77,118,98,126),(78,119,99,127),(79,120,100,128),(80,113,101,121)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 8A | ··· | 8H | 8I | ··· | 8AB | 8AC | ··· | 8AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | Q16 | C4○D4 | C8○D4 | C4○D8 | C8○D8 |
| kernel | C8×Q16 | C82 | Q8⋊C8 | C8⋊1C8 | C4×Q16 | C8×Q8 | Q8⋊C4 | C2.D8 | C2×Q16 | Q16 | C2×C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 16 | 2 | 4 | 2 | 4 | 4 | 8 |
Matrix representation of C8×Q16 ►in GL4(𝔽17) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 10 | 9 | 0 | 0 |
| 2 | 7 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 11 | 6 |
| 0 | 2 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 1 |
| 0 | 0 | 3 | 8 |
G:=sub<GL(4,GF(17))| [15,0,0,0,0,15,0,0,0,0,4,0,0,0,0,4],[10,2,0,0,9,7,0,0,0,0,0,11,0,0,3,6],[0,9,0,0,2,0,0,0,0,0,9,3,0,0,1,8] >;
C8×Q16 in GAP, Magma, Sage, TeX
C_8\times Q_{16} % in TeX
G:=Group("C8xQ16"); // GroupNames label
G:=SmallGroup(128,309);
// by ID
G=gap.SmallGroup(128,309);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,268,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations