p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊4Q16, C8.14D8, C82.5C2, C42.660C23, C4.4(C2×D8), C4.3(C2×Q16), (C2×C8).242D4, C8⋊2Q8.8C2, C2.7(C8⋊4D4), C4⋊Q16.6C2, C4⋊Q8.84C22, (C4×C8).395C22, C2.6(C4⋊Q16), C22.61(C4⋊1D4), (C2×C4).717(C2×D4), SmallGroup(128,445)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊4Q16
G = < a,b,c | a8=b8=1, c2=b4, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 208 in 96 conjugacy classes, 48 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C4×C8, C4×C8, C2.D8, C4⋊Q8, C2×Q16, C82, C4⋊Q16, C8⋊2Q8, C8⋊4Q16
Quotients: C1, C2, C22, D4, C23, D8, Q16, C2×D4, C4⋊1D4, C2×D8, C2×Q16, C8⋊4D4, C4⋊Q16, C8⋊4Q16
►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 35 28 95 42 85 101)(2 23 36 29 96 43 86 102)(3 24 37 30 89 44 87 103)(4 17 38 31 90 45 88 104)(5 18 39 32 91 46 81 97)(6 19 40 25 92 47 82 98)(7 20 33 26 93 48 83 99)(8 21 34 27 94 41 84 100)(9 62 123 51 71 111 74 114)(10 63 124 52 72 112 75 115)(11 64 125 53 65 105 76 116)(12 57 126 54 66 106 77 117)(13 58 127 55 67 107 78 118)(14 59 128 56 68 108 79 119)(15 60 121 49 69 109 80 120)(16 61 122 50 70 110 73 113)
(1 54 95 117)(2 53 96 116)(3 52 89 115)(4 51 90 114)(5 50 91 113)(6 49 92 120)(7 56 93 119)(8 55 94 118)(9 104 71 31)(10 103 72 30)(11 102 65 29)(12 101 66 28)(13 100 67 27)(14 99 68 26)(15 98 69 25)(16 97 70 32)(17 123 45 74)(18 122 46 73)(19 121 47 80)(20 128 48 79)(21 127 41 78)(22 126 42 77)(23 125 43 76)(24 124 44 75)(33 59 83 108)(34 58 84 107)(35 57 85 106)(36 64 86 105)(37 63 87 112)(38 62 88 111)(39 61 81 110)(40 60 82 109)
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), (1,22,35,28,95,42,85,101)(2,23,36,29,96,43,86,102)(3,24,37,30,89,44,87,103)(4,17,38,31,90,45,88,104)(5,18,39,32,91,46,81,97)(6,19,40,25,92,47,82,98)(7,20,33,26,93,48,83,99)(8,21,34,27,94,41,84,100)(9,62,123,51,71,111,74,114)(10,63,124,52,72,112,75,115)(11,64,125,53,65,105,76,116)(12,57,126,54,66,106,77,117)(13,58,127,55,67,107,78,118)(14,59,128,56,68,108,79,119)(15,60,121,49,69,109,80,120)(16,61,122,50,70,110,73,113), (1,54,95,117)(2,53,96,116)(3,52,89,115)(4,51,90,114)(5,50,91,113)(6,49,92,120)(7,56,93,119)(8,55,94,118)(9,104,71,31)(10,103,72,30)(11,102,65,29)(12,101,66,28)(13,100,67,27)(14,99,68,26)(15,98,69,25)(16,97,70,32)(17,123,45,74)(18,122,46,73)(19,121,47,80)(20,128,48,79)(21,127,41,78)(22,126,42,77)(23,125,43,76)(24,124,44,75)(33,59,83,108)(34,58,84,107)(35,57,85,106)(36,64,86,105)(37,63,87,112)(38,62,88,111)(39,61,81,110)(40,60,82,109)>;
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), (1,22,35,28,95,42,85,101)(2,23,36,29,96,43,86,102)(3,24,37,30,89,44,87,103)(4,17,38,31,90,45,88,104)(5,18,39,32,91,46,81,97)(6,19,40,25,92,47,82,98)(7,20,33,26,93,48,83,99)(8,21,34,27,94,41,84,100)(9,62,123,51,71,111,74,114)(10,63,124,52,72,112,75,115)(11,64,125,53,65,105,76,116)(12,57,126,54,66,106,77,117)(13,58,127,55,67,107,78,118)(14,59,128,56,68,108,79,119)(15,60,121,49,69,109,80,120)(16,61,122,50,70,110,73,113), (1,54,95,117)(2,53,96,116)(3,52,89,115)(4,51,90,114)(5,50,91,113)(6,49,92,120)(7,56,93,119)(8,55,94,118)(9,104,71,31)(10,103,72,30)(11,102,65,29)(12,101,66,28)(13,100,67,27)(14,99,68,26)(15,98,69,25)(16,97,70,32)(17,123,45,74)(18,122,46,73)(19,121,47,80)(20,128,48,79)(21,127,41,78)(22,126,42,77)(23,125,43,76)(24,124,44,75)(33,59,83,108)(34,58,84,107)(35,57,85,106)(36,64,86,105)(37,63,87,112)(38,62,88,111)(39,61,81,110)(40,60,82,109) );
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)], [(1,22,35,28,95,42,85,101),(2,23,36,29,96,43,86,102),(3,24,37,30,89,44,87,103),(4,17,38,31,90,45,88,104),(5,18,39,32,91,46,81,97),(6,19,40,25,92,47,82,98),(7,20,33,26,93,48,83,99),(8,21,34,27,94,41,84,100),(9,62,123,51,71,111,74,114),(10,63,124,52,72,112,75,115),(11,64,125,53,65,105,76,116),(12,57,126,54,66,106,77,117),(13,58,127,55,67,107,78,118),(14,59,128,56,68,108,79,119),(15,60,121,49,69,109,80,120),(16,61,122,50,70,110,73,113)], [(1,54,95,117),(2,53,96,116),(3,52,89,115),(4,51,90,114),(5,50,91,113),(6,49,92,120),(7,56,93,119),(8,55,94,118),(9,104,71,31),(10,103,72,30),(11,102,65,29),(12,101,66,28),(13,100,67,27),(14,99,68,26),(15,98,69,25),(16,97,70,32),(17,123,45,74),(18,122,46,73),(19,121,47,80),(20,128,48,79),(21,127,41,78),(22,126,42,77),(23,125,43,76),(24,124,44,75),(33,59,83,108),(34,58,84,107),(35,57,85,106),(36,64,86,105),(37,63,87,112),(38,62,88,111),(39,61,81,110),(40,60,82,109)]])
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 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | D4 | D8 | Q16 |
| kernel | C8⋊4Q16 | C82 | C4⋊Q16 | C8⋊2Q8 | C2×C8 | C8 | C8 |
| # reps | 1 | 1 | 2 | 4 | 6 | 8 | 16 |
Matrix representation of C8⋊4Q16 ►in GL4(𝔽17) generated by
| 3 | 3 | 0 | 0 |
| 14 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 14 | 3 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 11 | 11 |
| 0 | 0 | 3 | 0 |
| 1 | 10 | 0 | 0 |
| 10 | 16 | 0 | 0 |
| 0 | 0 | 9 | 15 |
| 0 | 0 | 7 | 8 |
G:=sub<GL(4,GF(17))| [3,14,0,0,3,3,0,0,0,0,1,0,0,0,0,1],[14,14,0,0,3,14,0,0,0,0,11,3,0,0,11,0],[1,10,0,0,10,16,0,0,0,0,9,7,0,0,15,8] >;
C8⋊4Q16 in GAP, Magma, Sage, TeX
C_8\rtimes_4Q_{16} % in TeX
G:=Group("C8:4Q16"); // GroupNames label
G:=SmallGroup(128,445);
// by ID
G=gap.SmallGroup(128,445);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,288,422,436,1123,136,2804,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=1,c^2=b^4,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations