direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×Q16, C8.C14, Q8.C14, C56.3C2, C14.16D4, C28.19C22, C2.5(C7×D4), C4.3(C2×C14), (C7×Q8).2C2, SmallGroup(112,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×Q16
G = < a,b,c | a7=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7×Q16
►Regular action on 112 points
Generators in S112
(1 82 101 107 37 31 61)(2 83 102 108 38 32 62)(3 84 103 109 39 25 63)(4 85 104 110 40 26 64)(5 86 97 111 33 27 57)(6 87 98 112 34 28 58)(7 88 99 105 35 29 59)(8 81 100 106 36 30 60)(9 43 80 17 51 93 72)(10 44 73 18 52 94 65)(11 45 74 19 53 95 66)(12 46 75 20 54 96 67)(13 47 76 21 55 89 68)(14 48 77 22 56 90 69)(15 41 78 23 49 91 70)(16 42 79 24 50 92 71)
(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)
(1 45 5 41)(2 44 6 48)(3 43 7 47)(4 42 8 46)(9 59 13 63)(10 58 14 62)(11 57 15 61)(12 64 16 60)(17 99 21 103)(18 98 22 102)(19 97 23 101)(20 104 24 100)(25 72 29 68)(26 71 30 67)(27 70 31 66)(28 69 32 65)(33 91 37 95)(34 90 38 94)(35 89 39 93)(36 96 40 92)(49 107 53 111)(50 106 54 110)(51 105 55 109)(52 112 56 108)(73 87 77 83)(74 86 78 82)(75 85 79 81)(76 84 80 88)
G:=sub<Sym(112)| (1,82,101,107,37,31,61)(2,83,102,108,38,32,62)(3,84,103,109,39,25,63)(4,85,104,110,40,26,64)(5,86,97,111,33,27,57)(6,87,98,112,34,28,58)(7,88,99,105,35,29,59)(8,81,100,106,36,30,60)(9,43,80,17,51,93,72)(10,44,73,18,52,94,65)(11,45,74,19,53,95,66)(12,46,75,20,54,96,67)(13,47,76,21,55,89,68)(14,48,77,22,56,90,69)(15,41,78,23,49,91,70)(16,42,79,24,50,92,71), (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), (1,45,5,41)(2,44,6,48)(3,43,7,47)(4,42,8,46)(9,59,13,63)(10,58,14,62)(11,57,15,61)(12,64,16,60)(17,99,21,103)(18,98,22,102)(19,97,23,101)(20,104,24,100)(25,72,29,68)(26,71,30,67)(27,70,31,66)(28,69,32,65)(33,91,37,95)(34,90,38,94)(35,89,39,93)(36,96,40,92)(49,107,53,111)(50,106,54,110)(51,105,55,109)(52,112,56,108)(73,87,77,83)(74,86,78,82)(75,85,79,81)(76,84,80,88)>;
G:=Group( (1,82,101,107,37,31,61)(2,83,102,108,38,32,62)(3,84,103,109,39,25,63)(4,85,104,110,40,26,64)(5,86,97,111,33,27,57)(6,87,98,112,34,28,58)(7,88,99,105,35,29,59)(8,81,100,106,36,30,60)(9,43,80,17,51,93,72)(10,44,73,18,52,94,65)(11,45,74,19,53,95,66)(12,46,75,20,54,96,67)(13,47,76,21,55,89,68)(14,48,77,22,56,90,69)(15,41,78,23,49,91,70)(16,42,79,24,50,92,71), (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), (1,45,5,41)(2,44,6,48)(3,43,7,47)(4,42,8,46)(9,59,13,63)(10,58,14,62)(11,57,15,61)(12,64,16,60)(17,99,21,103)(18,98,22,102)(19,97,23,101)(20,104,24,100)(25,72,29,68)(26,71,30,67)(27,70,31,66)(28,69,32,65)(33,91,37,95)(34,90,38,94)(35,89,39,93)(36,96,40,92)(49,107,53,111)(50,106,54,110)(51,105,55,109)(52,112,56,108)(73,87,77,83)(74,86,78,82)(75,85,79,81)(76,84,80,88) );
G=PermutationGroup([[(1,82,101,107,37,31,61),(2,83,102,108,38,32,62),(3,84,103,109,39,25,63),(4,85,104,110,40,26,64),(5,86,97,111,33,27,57),(6,87,98,112,34,28,58),(7,88,99,105,35,29,59),(8,81,100,106,36,30,60),(9,43,80,17,51,93,72),(10,44,73,18,52,94,65),(11,45,74,19,53,95,66),(12,46,75,20,54,96,67),(13,47,76,21,55,89,68),(14,48,77,22,56,90,69),(15,41,78,23,49,91,70),(16,42,79,24,50,92,71)], [(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)], [(1,45,5,41),(2,44,6,48),(3,43,7,47),(4,42,8,46),(9,59,13,63),(10,58,14,62),(11,57,15,61),(12,64,16,60),(17,99,21,103),(18,98,22,102),(19,97,23,101),(20,104,24,100),(25,72,29,68),(26,71,30,67),(27,70,31,66),(28,69,32,65),(33,91,37,95),(34,90,38,94),(35,89,39,93),(36,96,40,92),(49,107,53,111),(50,106,54,110),(51,105,55,109),(52,112,56,108),(73,87,77,83),(74,86,78,82),(75,85,79,81),(76,84,80,88)]])
C7×Q16 is a maximal subgroup of
C7⋊SD32 C7⋊Q32 Q16⋊D7 Q8.D14
49 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | D4 | Q16 | C7×D4 | C7×Q16 |
| kernel | C7×Q16 | C56 | C7×Q8 | Q16 | C8 | Q8 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 1 | 2 | 6 | 12 |
Matrix representation of C7×Q16 ►in GL2(𝔽113) generated by
| 30 | 0 |
| 0 | 30 |
| 31 | 82 |
| 31 | 31 |
| 97 | 46 |
| 46 | 16 |
G:=sub<GL(2,GF(113))| [30,0,0,30],[31,31,82,31],[97,46,46,16] >;
C7×Q16 in GAP, Magma, Sage, TeX
C_7\times Q_{16} % in TeX
G:=Group("C7xQ16"); // GroupNames label
G:=SmallGroup(112,26);
// by ID
G=gap.SmallGroup(112,26);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-2,280,301,286,1683,848,58]);
// Polycyclic
G:=Group<a,b,c|a^7=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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