p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊9Q16, Q8.1M4(2), C42.641C23, Q8⋊C8.2C2, C2.4(C4×Q16), C8⋊C8.7C2, (C2×C8).380D4, (C2×Q16).6C4, (C8×Q8).14C2, (C4×Q16).2C2, C4.58(C2×Q16), C2.D8.10C4, C8⋊1C8.13C2, C4.33(C8○D4), (C4×C8).39C22, Q8⋊C4.3C4, C8⋊4Q8.11C2, C4.131(C4○D8), C2.9(C8.26D4), C2.11(C8⋊9D4), C4⋊C8.223C22, C22.132(C4×D4), C4.27(C2×M4(2)), (C4×Q8).10C22, (C2×C8).31(C2×C4), C4⋊C4.135(C2×C4), (C2×C4).1477(C2×D4), (C2×Q8).136(C2×C4), (C2×C4).502(C4○D4), (C2×C4).333(C22×C4), SmallGroup(128,316)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊9Q16
G = < a,b,c | a8=b8=1, c2=b4, bab-1=a5, ac=ca, cbc-1=b-1 >
Subgroups: 120 in 76 conjugacy classes, 44 normal (40 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, C8⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C2.D8, C4×Q8, C2×Q16, C8⋊C8, Q8⋊C8, C8⋊1C8, C4×Q16, C8×Q8, C8⋊4Q8, C8⋊9Q16
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⋊9D4, C4×Q16, C8.26D4, C8⋊9Q16
►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)
(1 110 12 126 31 20 37 113)(2 107 13 123 32 17 38 118)(3 112 14 128 25 22 39 115)(4 109 15 125 26 19 40 120)(5 106 16 122 27 24 33 117)(6 111 9 127 28 21 34 114)(7 108 10 124 29 18 35 119)(8 105 11 121 30 23 36 116)(41 95 51 79 65 82 60 100)(42 92 52 76 66 87 61 97)(43 89 53 73 67 84 62 102)(44 94 54 78 68 81 63 99)(45 91 55 75 69 86 64 104)(46 96 56 80 70 83 57 101)(47 93 49 77 71 88 58 98)(48 90 50 74 72 85 59 103)
(1 63 31 54)(2 64 32 55)(3 57 25 56)(4 58 26 49)(5 59 27 50)(6 60 28 51)(7 61 29 52)(8 62 30 53)(9 65 34 41)(10 66 35 42)(11 67 36 43)(12 68 37 44)(13 69 38 45)(14 70 39 46)(15 71 40 47)(16 72 33 48)(17 91 107 86)(18 92 108 87)(19 93 109 88)(20 94 110 81)(21 95 111 82)(22 96 112 83)(23 89 105 84)(24 90 106 85)(73 116 102 121)(74 117 103 122)(75 118 104 123)(76 119 97 124)(77 120 98 125)(78 113 99 126)(79 114 100 127)(80 115 101 128)
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), (1,110,12,126,31,20,37,113)(2,107,13,123,32,17,38,118)(3,112,14,128,25,22,39,115)(4,109,15,125,26,19,40,120)(5,106,16,122,27,24,33,117)(6,111,9,127,28,21,34,114)(7,108,10,124,29,18,35,119)(8,105,11,121,30,23,36,116)(41,95,51,79,65,82,60,100)(42,92,52,76,66,87,61,97)(43,89,53,73,67,84,62,102)(44,94,54,78,68,81,63,99)(45,91,55,75,69,86,64,104)(46,96,56,80,70,83,57,101)(47,93,49,77,71,88,58,98)(48,90,50,74,72,85,59,103), (1,63,31,54)(2,64,32,55)(3,57,25,56)(4,58,26,49)(5,59,27,50)(6,60,28,51)(7,61,29,52)(8,62,30,53)(9,65,34,41)(10,66,35,42)(11,67,36,43)(12,68,37,44)(13,69,38,45)(14,70,39,46)(15,71,40,47)(16,72,33,48)(17,91,107,86)(18,92,108,87)(19,93,109,88)(20,94,110,81)(21,95,111,82)(22,96,112,83)(23,89,105,84)(24,90,106,85)(73,116,102,121)(74,117,103,122)(75,118,104,123)(76,119,97,124)(77,120,98,125)(78,113,99,126)(79,114,100,127)(80,115,101,128)>;
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), (1,110,12,126,31,20,37,113)(2,107,13,123,32,17,38,118)(3,112,14,128,25,22,39,115)(4,109,15,125,26,19,40,120)(5,106,16,122,27,24,33,117)(6,111,9,127,28,21,34,114)(7,108,10,124,29,18,35,119)(8,105,11,121,30,23,36,116)(41,95,51,79,65,82,60,100)(42,92,52,76,66,87,61,97)(43,89,53,73,67,84,62,102)(44,94,54,78,68,81,63,99)(45,91,55,75,69,86,64,104)(46,96,56,80,70,83,57,101)(47,93,49,77,71,88,58,98)(48,90,50,74,72,85,59,103), (1,63,31,54)(2,64,32,55)(3,57,25,56)(4,58,26,49)(5,59,27,50)(6,60,28,51)(7,61,29,52)(8,62,30,53)(9,65,34,41)(10,66,35,42)(11,67,36,43)(12,68,37,44)(13,69,38,45)(14,70,39,46)(15,71,40,47)(16,72,33,48)(17,91,107,86)(18,92,108,87)(19,93,109,88)(20,94,110,81)(21,95,111,82)(22,96,112,83)(23,89,105,84)(24,90,106,85)(73,116,102,121)(74,117,103,122)(75,118,104,123)(76,119,97,124)(77,120,98,125)(78,113,99,126)(79,114,100,127)(80,115,101,128) );
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)], [(1,110,12,126,31,20,37,113),(2,107,13,123,32,17,38,118),(3,112,14,128,25,22,39,115),(4,109,15,125,26,19,40,120),(5,106,16,122,27,24,33,117),(6,111,9,127,28,21,34,114),(7,108,10,124,29,18,35,119),(8,105,11,121,30,23,36,116),(41,95,51,79,65,82,60,100),(42,92,52,76,66,87,61,97),(43,89,53,73,67,84,62,102),(44,94,54,78,68,81,63,99),(45,91,55,75,69,86,64,104),(46,96,56,80,70,83,57,101),(47,93,49,77,71,88,58,98),(48,90,50,74,72,85,59,103)], [(1,63,31,54),(2,64,32,55),(3,57,25,56),(4,58,26,49),(5,59,27,50),(6,60,28,51),(7,61,29,52),(8,62,30,53),(9,65,34,41),(10,66,35,42),(11,67,36,43),(12,68,37,44),(13,69,38,45),(14,70,39,46),(15,71,40,47),(16,72,33,48),(17,91,107,86),(18,92,108,87),(19,93,109,88),(20,94,110,81),(21,95,111,82),(22,96,112,83),(23,89,105,84),(24,90,106,85),(73,116,102,121),(74,117,103,122),(75,118,104,123),(76,119,97,124),(77,120,98,125),(78,113,99,126),(79,114,100,127),(80,115,101,128)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | ··· | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q16 | C4○D4 | M4(2) | C8○D4 | C4○D8 | C8.26D4 |
| kernel | C8⋊9Q16 | C8⋊C8 | Q8⋊C8 | C8⋊1C8 | C4×Q16 | C8×Q8 | C8⋊4Q8 | Q8⋊C4 | C2.D8 | C2×Q16 | C2×C8 | C8 | C2×C4 | Q8 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 2 |
Matrix representation of C8⋊9Q16 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 13 | 0 |
| 0 | 6 | 0 | 0 |
| 14 | 6 | 0 | 0 |
| 0 | 0 | 12 | 11 |
| 0 | 0 | 10 | 5 |
| 8 | 15 | 0 | 0 |
| 7 | 9 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 9 | 0 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,0,13,0,0,1,0],[0,14,0,0,6,6,0,0,0,0,12,10,0,0,11,5],[8,7,0,0,15,9,0,0,0,0,0,9,0,0,2,0] >;
C8⋊9Q16 in GAP, Magma, Sage, TeX
C_8\rtimes_9Q_{16} % in TeX
G:=Group("C8:9Q16"); // GroupNames label
G:=SmallGroup(128,316);
// by ID
G=gap.SmallGroup(128,316);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,268,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=1,c^2=b^4,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations