p-group, metabelian, nilpotent (class 4), monomial
Aliases: C16⋊3Q8, C8.9Q16, C4.5SD32, C42.342D4, (C2×C4).87D8, C4.7(C4⋊Q8), C8.20(C2×Q8), (C4×C16).14C2, (C2×C8).256D4, C4.11(C2×Q16), C16⋊4C4.7C2, C8⋊2Q8.15C2, C2.8(C8⋊2Q8), C2.17(C2×SD32), (C2×C8).552C23, (C4×C8).405C22, C22.138(C2×D8), (C2×C16).100C22, C2.D8.31C22, (C2×C4).820(C2×D4), SmallGroup(128,986)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16⋊3Q8
G = < a,b,c | a16=b4=1, c2=b2, ab=ba, cac-1=a7, cbc-1=b-1 >
Subgroups: 152 in 64 conjugacy classes, 40 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C8, C2×C4, 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⋊4C4, C8⋊2Q8, C16⋊3Q8
Quotients: C1, C2, C22, D4, Q8, C23, D8, Q16, C2×D4, C2×Q8, SD32, C4⋊Q8, C2×D8, C2×Q16, C8⋊2Q8, C2×SD32, C16⋊3Q8
►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 40 60 27)(2 41 61 28)(3 42 62 29)(4 43 63 30)(5 44 64 31)(6 45 49 32)(7 46 50 17)(8 47 51 18)(9 48 52 19)(10 33 53 20)(11 34 54 21)(12 35 55 22)(13 36 56 23)(14 37 57 24)(15 38 58 25)(16 39 59 26)(65 122 105 93)(66 123 106 94)(67 124 107 95)(68 125 108 96)(69 126 109 81)(70 127 110 82)(71 128 111 83)(72 113 112 84)(73 114 97 85)(74 115 98 86)(75 116 99 87)(76 117 100 88)(77 118 101 89)(78 119 102 90)(79 120 103 91)(80 121 104 92)
(1 85 60 114)(2 92 61 121)(3 83 62 128)(4 90 63 119)(5 81 64 126)(6 88 49 117)(7 95 50 124)(8 86 51 115)(9 93 52 122)(10 84 53 113)(11 91 54 120)(12 82 55 127)(13 89 56 118)(14 96 57 125)(15 87 58 116)(16 94 59 123)(17 67 46 107)(18 74 47 98)(19 65 48 105)(20 72 33 112)(21 79 34 103)(22 70 35 110)(23 77 36 101)(24 68 37 108)(25 75 38 99)(26 66 39 106)(27 73 40 97)(28 80 41 104)(29 71 42 111)(30 78 43 102)(31 69 44 109)(32 76 45 100)
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), (1,40,60,27)(2,41,61,28)(3,42,62,29)(4,43,63,30)(5,44,64,31)(6,45,49,32)(7,46,50,17)(8,47,51,18)(9,48,52,19)(10,33,53,20)(11,34,54,21)(12,35,55,22)(13,36,56,23)(14,37,57,24)(15,38,58,25)(16,39,59,26)(65,122,105,93)(66,123,106,94)(67,124,107,95)(68,125,108,96)(69,126,109,81)(70,127,110,82)(71,128,111,83)(72,113,112,84)(73,114,97,85)(74,115,98,86)(75,116,99,87)(76,117,100,88)(77,118,101,89)(78,119,102,90)(79,120,103,91)(80,121,104,92), (1,85,60,114)(2,92,61,121)(3,83,62,128)(4,90,63,119)(5,81,64,126)(6,88,49,117)(7,95,50,124)(8,86,51,115)(9,93,52,122)(10,84,53,113)(11,91,54,120)(12,82,55,127)(13,89,56,118)(14,96,57,125)(15,87,58,116)(16,94,59,123)(17,67,46,107)(18,74,47,98)(19,65,48,105)(20,72,33,112)(21,79,34,103)(22,70,35,110)(23,77,36,101)(24,68,37,108)(25,75,38,99)(26,66,39,106)(27,73,40,97)(28,80,41,104)(29,71,42,111)(30,78,43,102)(31,69,44,109)(32,76,45,100)>;
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), (1,40,60,27)(2,41,61,28)(3,42,62,29)(4,43,63,30)(5,44,64,31)(6,45,49,32)(7,46,50,17)(8,47,51,18)(9,48,52,19)(10,33,53,20)(11,34,54,21)(12,35,55,22)(13,36,56,23)(14,37,57,24)(15,38,58,25)(16,39,59,26)(65,122,105,93)(66,123,106,94)(67,124,107,95)(68,125,108,96)(69,126,109,81)(70,127,110,82)(71,128,111,83)(72,113,112,84)(73,114,97,85)(74,115,98,86)(75,116,99,87)(76,117,100,88)(77,118,101,89)(78,119,102,90)(79,120,103,91)(80,121,104,92), (1,85,60,114)(2,92,61,121)(3,83,62,128)(4,90,63,119)(5,81,64,126)(6,88,49,117)(7,95,50,124)(8,86,51,115)(9,93,52,122)(10,84,53,113)(11,91,54,120)(12,82,55,127)(13,89,56,118)(14,96,57,125)(15,87,58,116)(16,94,59,123)(17,67,46,107)(18,74,47,98)(19,65,48,105)(20,72,33,112)(21,79,34,103)(22,70,35,110)(23,77,36,101)(24,68,37,108)(25,75,38,99)(26,66,39,106)(27,73,40,97)(28,80,41,104)(29,71,42,111)(30,78,43,102)(31,69,44,109)(32,76,45,100) );
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)], [(1,40,60,27),(2,41,61,28),(3,42,62,29),(4,43,63,30),(5,44,64,31),(6,45,49,32),(7,46,50,17),(8,47,51,18),(9,48,52,19),(10,33,53,20),(11,34,54,21),(12,35,55,22),(13,36,56,23),(14,37,57,24),(15,38,58,25),(16,39,59,26),(65,122,105,93),(66,123,106,94),(67,124,107,95),(68,125,108,96),(69,126,109,81),(70,127,110,82),(71,128,111,83),(72,113,112,84),(73,114,97,85),(74,115,98,86),(75,116,99,87),(76,117,100,88),(77,118,101,89),(78,119,102,90),(79,120,103,91),(80,121,104,92)], [(1,85,60,114),(2,92,61,121),(3,83,62,128),(4,90,63,119),(5,81,64,126),(6,88,49,117),(7,95,50,124),(8,86,51,115),(9,93,52,122),(10,84,53,113),(11,91,54,120),(12,82,55,127),(13,89,56,118),(14,96,57,125),(15,87,58,116),(16,94,59,123),(17,67,46,107),(18,74,47,98),(19,65,48,105),(20,72,33,112),(21,79,34,103),(22,70,35,110),(23,77,36,101),(24,68,37,108),(25,75,38,99),(26,66,39,106),(27,73,40,97),(28,80,41,104),(29,71,42,111),(30,78,43,102),(31,69,44,109),(32,76,45,100)]])
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 |
| type | + | + | + | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | Q8 | D4 | D4 | Q16 | D8 | SD32 |
| kernel | C16⋊3Q8 | C4×C16 | C16⋊4C4 | C8⋊2Q8 | C16 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 4 | 1 | 1 | 4 | 4 | 16 |
Matrix representation of C16⋊3Q8 ►in GL4(𝔽17) generated by
| 11 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 7 | 16 |
| 0 | 0 | 1 | 7 |
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 11 | 4 |
| 0 | 0 | 4 | 6 |
G:=sub<GL(4,GF(17))| [11,0,0,0,0,3,0,0,0,0,7,1,0,0,16,7],[4,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,1,0,0,0,0,0,11,4,0,0,4,6] >;
C16⋊3Q8 in GAP, Magma, Sage, TeX
C_{16}\rtimes_3Q_8 % in TeX
G:=Group("C16:3Q8"); // GroupNames label
G:=SmallGroup(128,986);
// by ID
G=gap.SmallGroup(128,986);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,64,422,604,1684,242,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^7,c*b*c^-1=b^-1>;
// generators/relations