p-group, metabelian, nilpotent (class 4), monomial
Aliases: C16⋊2Q8, C4.5D16, C4.4Q32, C8.8Q16, C42.340D4, (C2×C4).86D8, C4.5(C4⋊Q8), C8.18(C2×Q8), C4.9(C2×Q16), (C4×C16).10C2, (C2×C8).255D4, C2.11(C2×D16), C2.11(C2×Q32), C16⋊3C4.6C2, C8⋊2Q8.14C2, C2.6(C8⋊2Q8), (C4×C8).404C22, (C2×C8).550C23, (C2×C16).86C22, C22.136(C2×D8), C2.D8.29C22, (C2×C4).818(C2×D4), SmallGroup(128,984)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16⋊2Q8
G = < a,b,c | a16=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 152 in 64 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C4, C4, C4, C22, C8, C2×C4, C2×C4, Q8, C16, C42, C4⋊C4, C2×C8, C2×Q8, C4×C8, C2.D8, C2.D8, C2×C16, C4⋊Q8, C4×C16, C16⋊3C4, C8⋊2Q8, C16⋊2Q8
Quotients: C1, C2, C22, D4, Q8, C23, D8, Q16, C2×D4, C2×Q8, D16, Q32, C4⋊Q8, C2×D8, C2×Q16, C8⋊2Q8, C2×D16, C2×Q32, C16⋊2Q8
►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 27 97 128)(2 28 98 113)(3 29 99 114)(4 30 100 115)(5 31 101 116)(6 32 102 117)(7 17 103 118)(8 18 104 119)(9 19 105 120)(10 20 106 121)(11 21 107 122)(12 22 108 123)(13 23 109 124)(14 24 110 125)(15 25 111 126)(16 26 112 127)(33 77 84 50)(34 78 85 51)(35 79 86 52)(36 80 87 53)(37 65 88 54)(38 66 89 55)(39 67 90 56)(40 68 91 57)(41 69 92 58)(42 70 93 59)(43 71 94 60)(44 72 95 61)(45 73 96 62)(46 74 81 63)(47 75 82 64)(48 76 83 49)
(1 81 97 46)(2 96 98 45)(3 95 99 44)(4 94 100 43)(5 93 101 42)(6 92 102 41)(7 91 103 40)(8 90 104 39)(9 89 105 38)(10 88 106 37)(11 87 107 36)(12 86 108 35)(13 85 109 34)(14 84 110 33)(15 83 111 48)(16 82 112 47)(17 68 118 57)(18 67 119 56)(19 66 120 55)(20 65 121 54)(21 80 122 53)(22 79 123 52)(23 78 124 51)(24 77 125 50)(25 76 126 49)(26 75 127 64)(27 74 128 63)(28 73 113 62)(29 72 114 61)(30 71 115 60)(31 70 116 59)(32 69 117 58)
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), (1,27,97,128)(2,28,98,113)(3,29,99,114)(4,30,100,115)(5,31,101,116)(6,32,102,117)(7,17,103,118)(8,18,104,119)(9,19,105,120)(10,20,106,121)(11,21,107,122)(12,22,108,123)(13,23,109,124)(14,24,110,125)(15,25,111,126)(16,26,112,127)(33,77,84,50)(34,78,85,51)(35,79,86,52)(36,80,87,53)(37,65,88,54)(38,66,89,55)(39,67,90,56)(40,68,91,57)(41,69,92,58)(42,70,93,59)(43,71,94,60)(44,72,95,61)(45,73,96,62)(46,74,81,63)(47,75,82,64)(48,76,83,49), (1,81,97,46)(2,96,98,45)(3,95,99,44)(4,94,100,43)(5,93,101,42)(6,92,102,41)(7,91,103,40)(8,90,104,39)(9,89,105,38)(10,88,106,37)(11,87,107,36)(12,86,108,35)(13,85,109,34)(14,84,110,33)(15,83,111,48)(16,82,112,47)(17,68,118,57)(18,67,119,56)(19,66,120,55)(20,65,121,54)(21,80,122,53)(22,79,123,52)(23,78,124,51)(24,77,125,50)(25,76,126,49)(26,75,127,64)(27,74,128,63)(28,73,113,62)(29,72,114,61)(30,71,115,60)(31,70,116,59)(32,69,117,58)>;
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), (1,27,97,128)(2,28,98,113)(3,29,99,114)(4,30,100,115)(5,31,101,116)(6,32,102,117)(7,17,103,118)(8,18,104,119)(9,19,105,120)(10,20,106,121)(11,21,107,122)(12,22,108,123)(13,23,109,124)(14,24,110,125)(15,25,111,126)(16,26,112,127)(33,77,84,50)(34,78,85,51)(35,79,86,52)(36,80,87,53)(37,65,88,54)(38,66,89,55)(39,67,90,56)(40,68,91,57)(41,69,92,58)(42,70,93,59)(43,71,94,60)(44,72,95,61)(45,73,96,62)(46,74,81,63)(47,75,82,64)(48,76,83,49), (1,81,97,46)(2,96,98,45)(3,95,99,44)(4,94,100,43)(5,93,101,42)(6,92,102,41)(7,91,103,40)(8,90,104,39)(9,89,105,38)(10,88,106,37)(11,87,107,36)(12,86,108,35)(13,85,109,34)(14,84,110,33)(15,83,111,48)(16,82,112,47)(17,68,118,57)(18,67,119,56)(19,66,120,55)(20,65,121,54)(21,80,122,53)(22,79,123,52)(23,78,124,51)(24,77,125,50)(25,76,126,49)(26,75,127,64)(27,74,128,63)(28,73,113,62)(29,72,114,61)(30,71,115,60)(31,70,116,59)(32,69,117,58) );
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)], [(1,27,97,128),(2,28,98,113),(3,29,99,114),(4,30,100,115),(5,31,101,116),(6,32,102,117),(7,17,103,118),(8,18,104,119),(9,19,105,120),(10,20,106,121),(11,21,107,122),(12,22,108,123),(13,23,109,124),(14,24,110,125),(15,25,111,126),(16,26,112,127),(33,77,84,50),(34,78,85,51),(35,79,86,52),(36,80,87,53),(37,65,88,54),(38,66,89,55),(39,67,90,56),(40,68,91,57),(41,69,92,58),(42,70,93,59),(43,71,94,60),(44,72,95,61),(45,73,96,62),(46,74,81,63),(47,75,82,64),(48,76,83,49)], [(1,81,97,46),(2,96,98,45),(3,95,99,44),(4,94,100,43),(5,93,101,42),(6,92,102,41),(7,91,103,40),(8,90,104,39),(9,89,105,38),(10,88,106,37),(11,87,107,36),(12,86,108,35),(13,85,109,34),(14,84,110,33),(15,83,111,48),(16,82,112,47),(17,68,118,57),(18,67,119,56),(19,66,120,55),(20,65,121,54),(21,80,122,53),(22,79,123,52),(23,78,124,51),(24,77,125,50),(25,76,126,49),(26,75,127,64),(27,74,128,63),(28,73,113,62),(29,72,114,61),(30,71,115,60),(31,70,116,59),(32,69,117,58)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 16 | 16 | 16 | 16 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D4 | D4 | Q16 | D8 | D16 | Q32 |
| kernel | C16⋊2Q8 | C4×C16 | C16⋊3C4 | C8⋊2Q8 | C16 | C42 | C2×C8 | C8 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 4 | 1 | 1 | 4 | 4 | 8 | 8 |
Matrix representation of C16⋊2Q8 ►in GL4(𝔽17) generated by
| 4 | 11 | 0 | 0 |
| 6 | 4 | 0 | 0 |
| 0 | 0 | 13 | 6 |
| 0 | 0 | 11 | 13 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 12 | 0 | 0 |
| 12 | 5 | 0 | 0 |
| 0 | 0 | 11 | 4 |
| 0 | 0 | 4 | 6 |
G:=sub<GL(4,GF(17))| [4,6,0,0,11,4,0,0,0,0,13,11,0,0,6,13],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[12,12,0,0,12,5,0,0,0,0,11,4,0,0,4,6] >;
C16⋊2Q8 in GAP, Magma, Sage, TeX
C_{16}\rtimes_2Q_8 % in TeX
G:=Group("C16:2Q8"); // GroupNames label
G:=SmallGroup(128,984);
// by ID
G=gap.SmallGroup(128,984);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,64,422,268,1684,242,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations