p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊6Q16, C82.7C2, C8.8M4(2), C42.648C23, Q8⋊C8.9C2, C2.5(C4×Q16), (C2×C8).286D4, (C2×Q16).8C4, (C4×Q16).3C2, C4.59(C2×Q16), C2.D8.14C4, C8⋊1C8.14C2, C4.15(C8○D4), C2.12(C8○D8), C2.8(C8⋊6D4), Q8⋊C4.2C4, C8⋊4Q8.13C2, C4.9(C2×M4(2)), C4.134(C4○D8), C4⋊C8.226C22, (C4×C8).393C22, C22.139(C4×D4), (C4×Q8).14C22, C4⋊C4.61(C2×C4), (C2×C8).170(C2×C4), (C2×Q8).54(C2×C4), (C2×C4).1484(C2×D4), (C2×C4).509(C4○D4), (C2×C4).340(C22×C4), SmallGroup(128,323)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊6Q16
G = < a,b,c | a8=b8=1, c2=b4, ab=ba, cac-1=a5, cbc-1=b-1 >
Subgroups: 120 in 76 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C4, C4, C22, C8, C8, C2×C4, C2×C4, Q8, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C4×C8, C8⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C2.D8, C4×Q8, C2×Q16, C82, Q8⋊C8, C8⋊1C8, C4×Q16, C8⋊4Q8, C8⋊6Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, M4(2), Q16, C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C2×Q16, C4○D8, C8⋊6D4, C4×Q16, C8○D8, C8⋊6Q16
►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 22 34 117 25 111 10 122)(2 23 35 118 26 112 11 123)(3 24 36 119 27 105 12 124)(4 17 37 120 28 106 13 125)(5 18 38 113 29 107 14 126)(6 19 39 114 30 108 15 127)(7 20 40 115 31 109 16 128)(8 21 33 116 32 110 9 121)(41 94 56 76 70 83 59 98)(42 95 49 77 71 84 60 99)(43 96 50 78 72 85 61 100)(44 89 51 79 65 86 62 101)(45 90 52 80 66 87 63 102)(46 91 53 73 67 88 64 103)(47 92 54 74 68 81 57 104)(48 93 55 75 69 82 58 97)
(1 56 25 59)(2 53 26 64)(3 50 27 61)(4 55 28 58)(5 52 29 63)(6 49 30 60)(7 54 31 57)(8 51 32 62)(9 65 33 44)(10 70 34 41)(11 67 35 46)(12 72 36 43)(13 69 37 48)(14 66 38 45)(15 71 39 42)(16 68 40 47)(17 93 106 82)(18 90 107 87)(19 95 108 84)(20 92 109 81)(21 89 110 86)(22 94 111 83)(23 91 112 88)(24 96 105 85)(73 118 103 123)(74 115 104 128)(75 120 97 125)(76 117 98 122)(77 114 99 127)(78 119 100 124)(79 116 101 121)(80 113 102 126)
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), (1,22,34,117,25,111,10,122)(2,23,35,118,26,112,11,123)(3,24,36,119,27,105,12,124)(4,17,37,120,28,106,13,125)(5,18,38,113,29,107,14,126)(6,19,39,114,30,108,15,127)(7,20,40,115,31,109,16,128)(8,21,33,116,32,110,9,121)(41,94,56,76,70,83,59,98)(42,95,49,77,71,84,60,99)(43,96,50,78,72,85,61,100)(44,89,51,79,65,86,62,101)(45,90,52,80,66,87,63,102)(46,91,53,73,67,88,64,103)(47,92,54,74,68,81,57,104)(48,93,55,75,69,82,58,97), (1,56,25,59)(2,53,26,64)(3,50,27,61)(4,55,28,58)(5,52,29,63)(6,49,30,60)(7,54,31,57)(8,51,32,62)(9,65,33,44)(10,70,34,41)(11,67,35,46)(12,72,36,43)(13,69,37,48)(14,66,38,45)(15,71,39,42)(16,68,40,47)(17,93,106,82)(18,90,107,87)(19,95,108,84)(20,92,109,81)(21,89,110,86)(22,94,111,83)(23,91,112,88)(24,96,105,85)(73,118,103,123)(74,115,104,128)(75,120,97,125)(76,117,98,122)(77,114,99,127)(78,119,100,124)(79,116,101,121)(80,113,102,126)>;
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), (1,22,34,117,25,111,10,122)(2,23,35,118,26,112,11,123)(3,24,36,119,27,105,12,124)(4,17,37,120,28,106,13,125)(5,18,38,113,29,107,14,126)(6,19,39,114,30,108,15,127)(7,20,40,115,31,109,16,128)(8,21,33,116,32,110,9,121)(41,94,56,76,70,83,59,98)(42,95,49,77,71,84,60,99)(43,96,50,78,72,85,61,100)(44,89,51,79,65,86,62,101)(45,90,52,80,66,87,63,102)(46,91,53,73,67,88,64,103)(47,92,54,74,68,81,57,104)(48,93,55,75,69,82,58,97), (1,56,25,59)(2,53,26,64)(3,50,27,61)(4,55,28,58)(5,52,29,63)(6,49,30,60)(7,54,31,57)(8,51,32,62)(9,65,33,44)(10,70,34,41)(11,67,35,46)(12,72,36,43)(13,69,37,48)(14,66,38,45)(15,71,39,42)(16,68,40,47)(17,93,106,82)(18,90,107,87)(19,95,108,84)(20,92,109,81)(21,89,110,86)(22,94,111,83)(23,91,112,88)(24,96,105,85)(73,118,103,123)(74,115,104,128)(75,120,97,125)(76,117,98,122)(77,114,99,127)(78,119,100,124)(79,116,101,121)(80,113,102,126) );
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)], [(1,22,34,117,25,111,10,122),(2,23,35,118,26,112,11,123),(3,24,36,119,27,105,12,124),(4,17,37,120,28,106,13,125),(5,18,38,113,29,107,14,126),(6,19,39,114,30,108,15,127),(7,20,40,115,31,109,16,128),(8,21,33,116,32,110,9,121),(41,94,56,76,70,83,59,98),(42,95,49,77,71,84,60,99),(43,96,50,78,72,85,61,100),(44,89,51,79,65,86,62,101),(45,90,52,80,66,87,63,102),(46,91,53,73,67,88,64,103),(47,92,54,74,68,81,57,104),(48,93,55,75,69,82,58,97)], [(1,56,25,59),(2,53,26,64),(3,50,27,61),(4,55,28,58),(5,52,29,63),(6,49,30,60),(7,54,31,57),(8,51,32,62),(9,65,33,44),(10,70,34,41),(11,67,35,46),(12,72,36,43),(13,69,37,48),(14,66,38,45),(15,71,39,42),(16,68,40,47),(17,93,106,82),(18,90,107,87),(19,95,108,84),(20,92,109,81),(21,89,110,86),(22,94,111,83),(23,91,112,88),(24,96,105,85),(73,118,103,123),(74,115,104,128),(75,120,97,125),(76,117,98,122),(77,114,99,127),(78,119,100,124),(79,116,101,121),(80,113,102,126)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8X | 8Y | 8Z | 8AA | 8AB |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 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 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | M4(2) | Q16 | C4○D4 | C8○D4 | C4○D8 | C8○D8 |
| kernel | C8⋊6Q16 | C82 | Q8⋊C8 | C8⋊1C8 | C4×Q16 | C8⋊4Q8 | Q8⋊C4 | C2.D8 | C2×Q16 | C2×C8 | C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 8 |
Matrix representation of C8⋊6Q16 ►in GL4(𝔽17) generated by
| 0 | 9 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 14 | 14 | 0 | 0 |
| 3 | 14 | 0 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 9 | 0 |
| 1 | 7 | 0 | 0 |
| 7 | 16 | 0 | 0 |
| 0 | 0 | 11 | 8 |
| 0 | 0 | 2 | 6 |
G:=sub<GL(4,GF(17))| [0,8,0,0,9,0,0,0,0,0,0,4,0,0,1,0],[14,3,0,0,14,14,0,0,0,0,0,9,0,0,15,0],[1,7,0,0,7,16,0,0,0,0,11,2,0,0,8,6] >;
C8⋊6Q16 in GAP, Magma, Sage, TeX
C_8\rtimes_6Q_{16} % in TeX
G:=Group("C8:6Q16"); // GroupNames label
G:=SmallGroup(128,323);
// by ID
G=gap.SmallGroup(128,323);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,723,268,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=1,c^2=b^4,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^-1>;
// generators/relations