p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊5Q16, C82.6C2, C8.17SD16, C42.654C23, C4.1(C2×Q16), C4.2(C4○D8), (C2×C8).221D4, C8⋊3Q8.6C2, C8⋊2Q8.6C2, C4.3(C2×SD16), C2.7(C8⋊5D4), C4⋊Q16.5C2, C4⋊Q8.79C22, (C4×C8).428C22, C2.4(C4⋊Q16), C4.SD16.3C2, C2.7(C8.12D4), C22.55(C4⋊1D4), (C2×C4).711(C2×D4), SmallGroup(128,439)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊5Q16
G = < a,b,c | a8=b8=1, c2=b4, ab=ba, cac-1=a3, cbc-1=b-1 >
Subgroups: 192 in 90 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C4, C4, C22, C8, C8, C2×C4, C2×C4, Q8, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C4×C8, Q8⋊C4, C4.Q8, C2.D8, C4⋊Q8, C2×Q16, C82, C4.SD16, C4⋊Q16, C8⋊3Q8, C8⋊2Q8, C8⋊5Q16
Quotients: C1, C2, C22, D4, C23, SD16, Q16, C2×D4, C4⋊1D4, C2×SD16, C2×Q16, C4○D8, C8⋊5D4, C4⋊Q16, C8.12D4, C8⋊5Q16
►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 63 23 73 47 38 16 54)(2 64 24 74 48 39 9 55)(3 57 17 75 41 40 10 56)(4 58 18 76 42 33 11 49)(5 59 19 77 43 34 12 50)(6 60 20 78 44 35 13 51)(7 61 21 79 45 36 14 52)(8 62 22 80 46 37 15 53)(25 91 123 88 120 72 104 109)(26 92 124 81 113 65 97 110)(27 93 125 82 114 66 98 111)(28 94 126 83 115 67 99 112)(29 95 127 84 116 68 100 105)(30 96 128 85 117 69 101 106)(31 89 121 86 118 70 102 107)(32 90 122 87 119 71 103 108)
(1 111 47 82)(2 106 48 85)(3 109 41 88)(4 112 42 83)(5 107 43 86)(6 110 44 81)(7 105 45 84)(8 108 46 87)(9 96 24 69)(10 91 17 72)(11 94 18 67)(12 89 19 70)(13 92 20 65)(14 95 21 68)(15 90 22 71)(16 93 23 66)(25 75 120 56)(26 78 113 51)(27 73 114 54)(28 76 115 49)(29 79 116 52)(30 74 117 55)(31 77 118 50)(32 80 119 53)(33 126 58 99)(34 121 59 102)(35 124 60 97)(36 127 61 100)(37 122 62 103)(38 125 63 98)(39 128 64 101)(40 123 57 104)
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), (1,63,23,73,47,38,16,54)(2,64,24,74,48,39,9,55)(3,57,17,75,41,40,10,56)(4,58,18,76,42,33,11,49)(5,59,19,77,43,34,12,50)(6,60,20,78,44,35,13,51)(7,61,21,79,45,36,14,52)(8,62,22,80,46,37,15,53)(25,91,123,88,120,72,104,109)(26,92,124,81,113,65,97,110)(27,93,125,82,114,66,98,111)(28,94,126,83,115,67,99,112)(29,95,127,84,116,68,100,105)(30,96,128,85,117,69,101,106)(31,89,121,86,118,70,102,107)(32,90,122,87,119,71,103,108), (1,111,47,82)(2,106,48,85)(3,109,41,88)(4,112,42,83)(5,107,43,86)(6,110,44,81)(7,105,45,84)(8,108,46,87)(9,96,24,69)(10,91,17,72)(11,94,18,67)(12,89,19,70)(13,92,20,65)(14,95,21,68)(15,90,22,71)(16,93,23,66)(25,75,120,56)(26,78,113,51)(27,73,114,54)(28,76,115,49)(29,79,116,52)(30,74,117,55)(31,77,118,50)(32,80,119,53)(33,126,58,99)(34,121,59,102)(35,124,60,97)(36,127,61,100)(37,122,62,103)(38,125,63,98)(39,128,64,101)(40,123,57,104)>;
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), (1,63,23,73,47,38,16,54)(2,64,24,74,48,39,9,55)(3,57,17,75,41,40,10,56)(4,58,18,76,42,33,11,49)(5,59,19,77,43,34,12,50)(6,60,20,78,44,35,13,51)(7,61,21,79,45,36,14,52)(8,62,22,80,46,37,15,53)(25,91,123,88,120,72,104,109)(26,92,124,81,113,65,97,110)(27,93,125,82,114,66,98,111)(28,94,126,83,115,67,99,112)(29,95,127,84,116,68,100,105)(30,96,128,85,117,69,101,106)(31,89,121,86,118,70,102,107)(32,90,122,87,119,71,103,108), (1,111,47,82)(2,106,48,85)(3,109,41,88)(4,112,42,83)(5,107,43,86)(6,110,44,81)(7,105,45,84)(8,108,46,87)(9,96,24,69)(10,91,17,72)(11,94,18,67)(12,89,19,70)(13,92,20,65)(14,95,21,68)(15,90,22,71)(16,93,23,66)(25,75,120,56)(26,78,113,51)(27,73,114,54)(28,76,115,49)(29,79,116,52)(30,74,117,55)(31,77,118,50)(32,80,119,53)(33,126,58,99)(34,121,59,102)(35,124,60,97)(36,127,61,100)(37,122,62,103)(38,125,63,98)(39,128,64,101)(40,123,57,104) );
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)], [(1,63,23,73,47,38,16,54),(2,64,24,74,48,39,9,55),(3,57,17,75,41,40,10,56),(4,58,18,76,42,33,11,49),(5,59,19,77,43,34,12,50),(6,60,20,78,44,35,13,51),(7,61,21,79,45,36,14,52),(8,62,22,80,46,37,15,53),(25,91,123,88,120,72,104,109),(26,92,124,81,113,65,97,110),(27,93,125,82,114,66,98,111),(28,94,126,83,115,67,99,112),(29,95,127,84,116,68,100,105),(30,96,128,85,117,69,101,106),(31,89,121,86,118,70,102,107),(32,90,122,87,119,71,103,108)], [(1,111,47,82),(2,106,48,85),(3,109,41,88),(4,112,42,83),(5,107,43,86),(6,110,44,81),(7,105,45,84),(8,108,46,87),(9,96,24,69),(10,91,17,72),(11,94,18,67),(12,89,19,70),(13,92,20,65),(14,95,21,68),(15,90,22,71),(16,93,23,66),(25,75,120,56),(26,78,113,51),(27,73,114,54),(28,76,115,49),(29,79,116,52),(30,74,117,55),(31,77,118,50),(32,80,119,53),(33,126,58,99),(34,121,59,102),(35,124,60,97),(36,127,61,100),(37,122,62,103),(38,125,63,98),(39,128,64,101),(40,123,57,104)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 16 | 16 | 16 | 16 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | SD16 | Q16 | C4○D8 |
| kernel | C8⋊5Q16 | C82 | C4.SD16 | C4⋊Q16 | C8⋊3Q8 | C8⋊2Q8 | C2×C8 | C8 | C8 | C4 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 6 | 8 | 8 | 8 |
Matrix representation of C8⋊5Q16 ►in GL4(𝔽17) generated by
| 8 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 5 |
| 0 | 0 | 0 | 8 |
| 15 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 12 | 4 |
| 0 | 0 | 11 | 5 |
G:=sub<GL(4,GF(17))| [8,0,0,0,0,2,0,0,0,0,2,0,0,0,5,8],[15,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,1,0,0,0,0,0,12,11,0,0,4,5] >;
C8⋊5Q16 in GAP, Magma, Sage, TeX
C_8\rtimes_5Q_{16} % in TeX
G:=Group("C8:5Q16"); // GroupNames label
G:=SmallGroup(128,439);
// by ID
G=gap.SmallGroup(128,439);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,288,422,100,1123,136,2804,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=1,c^2=b^4,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=b^-1>;
// generators/relations