p-group, metabelian, nilpotent (class 4), monomial
Aliases: C16.5Q8, C8.11Q16, C42.341D4, (C2×C4).64D8, C4.6(C4⋊Q8), C8.19(C2×Q8), (C4×C16).12C2, (C2×C8).281D4, C4.10(C2×Q16), C16⋊4C4.6C2, C16⋊3C4.7C2, C2.7(C8⋊2Q8), C8.5Q8.3C2, C2.19(C4○D16), (C4×C8).415C22, (C2×C8).551C23, (C2×C16).87C22, C22.137(C2×D8), C2.D8.30C22, (C2×C4).819(C2×D4), SmallGroup(128,985)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16.5Q8
G = < a,b,c | a16=b4=1, c2=a8b2, ab=ba, cac-1=a7, cbc-1=a8b-1 >
Smallest permutation representation of C16.5Q8
►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 93 28 46)(2 94 29 47)(3 95 30 48)(4 96 31 33)(5 81 32 34)(6 82 17 35)(7 83 18 36)(8 84 19 37)(9 85 20 38)(10 86 21 39)(11 87 22 40)(12 88 23 41)(13 89 24 42)(14 90 25 43)(15 91 26 44)(16 92 27 45)(49 68 126 111)(50 69 127 112)(51 70 128 97)(52 71 113 98)(53 72 114 99)(54 73 115 100)(55 74 116 101)(56 75 117 102)(57 76 118 103)(58 77 119 104)(59 78 120 105)(60 79 121 106)(61 80 122 107)(62 65 123 108)(63 66 124 109)(64 67 125 110)
(1 79 20 98)(2 70 21 105)(3 77 22 112)(4 68 23 103)(5 75 24 110)(6 66 25 101)(7 73 26 108)(8 80 27 99)(9 71 28 106)(10 78 29 97)(11 69 30 104)(12 76 31 111)(13 67 32 102)(14 74 17 109)(15 65 18 100)(16 72 19 107)(33 118 88 49)(34 125 89 56)(35 116 90 63)(36 123 91 54)(37 114 92 61)(38 121 93 52)(39 128 94 59)(40 119 95 50)(41 126 96 57)(42 117 81 64)(43 124 82 55)(44 115 83 62)(45 122 84 53)(46 113 85 60)(47 120 86 51)(48 127 87 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), (1,93,28,46)(2,94,29,47)(3,95,30,48)(4,96,31,33)(5,81,32,34)(6,82,17,35)(7,83,18,36)(8,84,19,37)(9,85,20,38)(10,86,21,39)(11,87,22,40)(12,88,23,41)(13,89,24,42)(14,90,25,43)(15,91,26,44)(16,92,27,45)(49,68,126,111)(50,69,127,112)(51,70,128,97)(52,71,113,98)(53,72,114,99)(54,73,115,100)(55,74,116,101)(56,75,117,102)(57,76,118,103)(58,77,119,104)(59,78,120,105)(60,79,121,106)(61,80,122,107)(62,65,123,108)(63,66,124,109)(64,67,125,110), (1,79,20,98)(2,70,21,105)(3,77,22,112)(4,68,23,103)(5,75,24,110)(6,66,25,101)(7,73,26,108)(8,80,27,99)(9,71,28,106)(10,78,29,97)(11,69,30,104)(12,76,31,111)(13,67,32,102)(14,74,17,109)(15,65,18,100)(16,72,19,107)(33,118,88,49)(34,125,89,56)(35,116,90,63)(36,123,91,54)(37,114,92,61)(38,121,93,52)(39,128,94,59)(40,119,95,50)(41,126,96,57)(42,117,81,64)(43,124,82,55)(44,115,83,62)(45,122,84,53)(46,113,85,60)(47,120,86,51)(48,127,87,58)>;
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), (1,93,28,46)(2,94,29,47)(3,95,30,48)(4,96,31,33)(5,81,32,34)(6,82,17,35)(7,83,18,36)(8,84,19,37)(9,85,20,38)(10,86,21,39)(11,87,22,40)(12,88,23,41)(13,89,24,42)(14,90,25,43)(15,91,26,44)(16,92,27,45)(49,68,126,111)(50,69,127,112)(51,70,128,97)(52,71,113,98)(53,72,114,99)(54,73,115,100)(55,74,116,101)(56,75,117,102)(57,76,118,103)(58,77,119,104)(59,78,120,105)(60,79,121,106)(61,80,122,107)(62,65,123,108)(63,66,124,109)(64,67,125,110), (1,79,20,98)(2,70,21,105)(3,77,22,112)(4,68,23,103)(5,75,24,110)(6,66,25,101)(7,73,26,108)(8,80,27,99)(9,71,28,106)(10,78,29,97)(11,69,30,104)(12,76,31,111)(13,67,32,102)(14,74,17,109)(15,65,18,100)(16,72,19,107)(33,118,88,49)(34,125,89,56)(35,116,90,63)(36,123,91,54)(37,114,92,61)(38,121,93,52)(39,128,94,59)(40,119,95,50)(41,126,96,57)(42,117,81,64)(43,124,82,55)(44,115,83,62)(45,122,84,53)(46,113,85,60)(47,120,86,51)(48,127,87,58) );
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)], [(1,93,28,46),(2,94,29,47),(3,95,30,48),(4,96,31,33),(5,81,32,34),(6,82,17,35),(7,83,18,36),(8,84,19,37),(9,85,20,38),(10,86,21,39),(11,87,22,40),(12,88,23,41),(13,89,24,42),(14,90,25,43),(15,91,26,44),(16,92,27,45),(49,68,126,111),(50,69,127,112),(51,70,128,97),(52,71,113,98),(53,72,114,99),(54,73,115,100),(55,74,116,101),(56,75,117,102),(57,76,118,103),(58,77,119,104),(59,78,120,105),(60,79,121,106),(61,80,122,107),(62,65,123,108),(63,66,124,109),(64,67,125,110)], [(1,79,20,98),(2,70,21,105),(3,77,22,112),(4,68,23,103),(5,75,24,110),(6,66,25,101),(7,73,26,108),(8,80,27,99),(9,71,28,106),(10,78,29,97),(11,69,30,104),(12,76,31,111),(13,67,32,102),(14,74,17,109),(15,65,18,100),(16,72,19,107),(33,118,88,49),(34,125,89,56),(35,116,90,63),(36,123,91,54),(37,114,92,61),(38,121,93,52),(39,128,94,59),(40,119,95,50),(41,126,96,57),(42,117,81,64),(43,124,82,55),(44,115,83,62),(45,122,84,53),(46,113,85,60),(47,120,86,51),(48,127,87,58)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D4 | D4 | Q16 | D8 | C4○D16 |
| kernel | C16.5Q8 | C4×C16 | C16⋊3C4 | C16⋊4C4 | C8.5Q8 | C16 | C42 | C2×C8 | C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 4 | 4 | 16 |
Matrix representation of C16.5Q8 ►in GL4(𝔽17) generated by
| 16 | 10 | 0 | 0 |
| 7 | 16 | 0 | 0 |
| 0 | 0 | 10 | 1 |
| 0 | 0 | 16 | 10 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
| 11 | 13 | 0 | 0 |
| 13 | 6 | 0 | 0 |
| 0 | 0 | 12 | 5 |
| 0 | 0 | 5 | 5 |
G:=sub<GL(4,GF(17))| [16,7,0,0,10,16,0,0,0,0,10,16,0,0,1,10],[4,0,0,0,0,4,0,0,0,0,0,4,0,0,13,0],[11,13,0,0,13,6,0,0,0,0,12,5,0,0,5,5] >;
C16.5Q8 in GAP, Magma, Sage, TeX
C_{16}._5Q_8 % in TeX
G:=Group("C16.5Q8"); // GroupNames label
G:=SmallGroup(128,985);
// by ID
G=gap.SmallGroup(128,985);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,512,422,268,1684,242,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^4=1,c^2=a^8*b^2,a*b=b*a,c*a*c^-1=a^7,c*b*c^-1=a^8*b^-1>;
// generators/relations
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