p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8.7Q16, C82.4C2, C42.657C23, C4.2(C2×Q16), C4.5(C4○D8), (C2×C8).224D4, C8⋊2Q8.7C2, C4⋊Q8.82C22, (C4×C8).370C22, C2.5(C4⋊Q16), C4.SD16.4C2, C2.10(C8.12D4), C22.58(C4⋊1D4), (C2×C4).714(C2×D4), SmallGroup(128,442)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.7Q16
G = < a,b,c | a8=b8=1, c2=b4, ab=ba, cac-1=a-1, cbc-1=a4b-1 >
Subgroups: 176 in 84 conjugacy classes, 40 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, C42, C4⋊C4, C2×C8, C2×Q8, C4×C8, C4×C8, Q8⋊C4, C2.D8, C4⋊Q8, C82, C4.SD16, C8⋊2Q8, C8.7Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C4⋊1D4, C2×Q16, C4○D8, C4⋊Q16, C8.12D4, C8.7Q16
►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 21 34 27 96 42 86 97)(2 22 35 28 89 43 87 98)(3 23 36 29 90 44 88 99)(4 24 37 30 91 45 81 100)(5 17 38 31 92 46 82 101)(6 18 39 32 93 47 83 102)(7 19 40 25 94 48 84 103)(8 20 33 26 95 41 85 104)(9 58 127 51 67 116 74 107)(10 59 128 52 68 117 75 108)(11 60 121 53 69 118 76 109)(12 61 122 54 70 119 77 110)(13 62 123 55 71 120 78 111)(14 63 124 56 72 113 79 112)(15 64 125 49 65 114 80 105)(16 57 126 50 66 115 73 106)
(1 54 96 110)(2 53 89 109)(3 52 90 108)(4 51 91 107)(5 50 92 106)(6 49 93 105)(7 56 94 112)(8 55 95 111)(9 104 67 26)(10 103 68 25)(11 102 69 32)(12 101 70 31)(13 100 71 30)(14 99 72 29)(15 98 65 28)(16 97 66 27)(17 122 46 77)(18 121 47 76)(19 128 48 75)(20 127 41 74)(21 126 42 73)(22 125 43 80)(23 124 44 79)(24 123 45 78)(33 62 85 120)(34 61 86 119)(35 60 87 118)(36 59 88 117)(37 58 81 116)(38 57 82 115)(39 64 83 114)(40 63 84 113)
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), (1,21,34,27,96,42,86,97)(2,22,35,28,89,43,87,98)(3,23,36,29,90,44,88,99)(4,24,37,30,91,45,81,100)(5,17,38,31,92,46,82,101)(6,18,39,32,93,47,83,102)(7,19,40,25,94,48,84,103)(8,20,33,26,95,41,85,104)(9,58,127,51,67,116,74,107)(10,59,128,52,68,117,75,108)(11,60,121,53,69,118,76,109)(12,61,122,54,70,119,77,110)(13,62,123,55,71,120,78,111)(14,63,124,56,72,113,79,112)(15,64,125,49,65,114,80,105)(16,57,126,50,66,115,73,106), (1,54,96,110)(2,53,89,109)(3,52,90,108)(4,51,91,107)(5,50,92,106)(6,49,93,105)(7,56,94,112)(8,55,95,111)(9,104,67,26)(10,103,68,25)(11,102,69,32)(12,101,70,31)(13,100,71,30)(14,99,72,29)(15,98,65,28)(16,97,66,27)(17,122,46,77)(18,121,47,76)(19,128,48,75)(20,127,41,74)(21,126,42,73)(22,125,43,80)(23,124,44,79)(24,123,45,78)(33,62,85,120)(34,61,86,119)(35,60,87,118)(36,59,88,117)(37,58,81,116)(38,57,82,115)(39,64,83,114)(40,63,84,113)>;
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), (1,21,34,27,96,42,86,97)(2,22,35,28,89,43,87,98)(3,23,36,29,90,44,88,99)(4,24,37,30,91,45,81,100)(5,17,38,31,92,46,82,101)(6,18,39,32,93,47,83,102)(7,19,40,25,94,48,84,103)(8,20,33,26,95,41,85,104)(9,58,127,51,67,116,74,107)(10,59,128,52,68,117,75,108)(11,60,121,53,69,118,76,109)(12,61,122,54,70,119,77,110)(13,62,123,55,71,120,78,111)(14,63,124,56,72,113,79,112)(15,64,125,49,65,114,80,105)(16,57,126,50,66,115,73,106), (1,54,96,110)(2,53,89,109)(3,52,90,108)(4,51,91,107)(5,50,92,106)(6,49,93,105)(7,56,94,112)(8,55,95,111)(9,104,67,26)(10,103,68,25)(11,102,69,32)(12,101,70,31)(13,100,71,30)(14,99,72,29)(15,98,65,28)(16,97,66,27)(17,122,46,77)(18,121,47,76)(19,128,48,75)(20,127,41,74)(21,126,42,73)(22,125,43,80)(23,124,44,79)(24,123,45,78)(33,62,85,120)(34,61,86,119)(35,60,87,118)(36,59,88,117)(37,58,81,116)(38,57,82,115)(39,64,83,114)(40,63,84,113) );
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)], [(1,21,34,27,96,42,86,97),(2,22,35,28,89,43,87,98),(3,23,36,29,90,44,88,99),(4,24,37,30,91,45,81,100),(5,17,38,31,92,46,82,101),(6,18,39,32,93,47,83,102),(7,19,40,25,94,48,84,103),(8,20,33,26,95,41,85,104),(9,58,127,51,67,116,74,107),(10,59,128,52,68,117,75,108),(11,60,121,53,69,118,76,109),(12,61,122,54,70,119,77,110),(13,62,123,55,71,120,78,111),(14,63,124,56,72,113,79,112),(15,64,125,49,65,114,80,105),(16,57,126,50,66,115,73,106)], [(1,54,96,110),(2,53,89,109),(3,52,90,108),(4,51,91,107),(5,50,92,106),(6,49,93,105),(7,56,94,112),(8,55,95,111),(9,104,67,26),(10,103,68,25),(11,102,69,32),(12,101,70,31),(13,100,71,30),(14,99,72,29),(15,98,65,28),(16,97,66,27),(17,122,46,77),(18,121,47,76),(19,128,48,75),(20,127,41,74),(21,126,42,73),(22,125,43,80),(23,124,44,79),(24,123,45,78),(33,62,85,120),(34,61,86,119),(35,60,87,118),(36,59,88,117),(37,58,81,116),(38,57,82,115),(39,64,83,114),(40,63,84,113)]])
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 | Q16 | C4○D8 |
| kernel | C8.7Q16 | C82 | C4.SD16 | C8⋊2Q8 | C2×C8 | C8 | C4 |
| # reps | 1 | 1 | 4 | 2 | 6 | 8 | 16 |
Matrix representation of C8.7Q16 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 2 |
| 2 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,4,0,0,0,0,9,0,0,0,0,2],[2,0,0,0,0,9,0,0,0,0,1,0,0,0,0,16],[0,16,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C8.7Q16 in GAP, Magma, Sage, TeX
C_8._7Q_{16} % in TeX
G:=Group("C8.7Q16"); // GroupNames label
G:=SmallGroup(128,442);
// by ID
G=gap.SmallGroup(128,442);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,512,422,268,1123,136,2804,172]);
// 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=a^4*b^-1>;
// generators/relations