direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C7xD16, C112:3C2, C16:1C14, D8:1C14, C14.15D8, C28.36D4, C56.24C22, (C7xD8):5C2, C4.1(C7xD4), C2.3(C7xD8), C8.2(C2xC14), SmallGroup(224,60)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7xD16
G = < a,b,c | a7=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C22, C7, D4, C14, D8, C2xC14, D16, C7xD4, C7xD8, C7xD16
Smallest permutation representation of C7xD16
►On 112 points
Generators in S112
(1 33 54 69 90 28 108)(2 34 55 70 91 29 109)(3 35 56 71 92 30 110)(4 36 57 72 93 31 111)(5 37 58 73 94 32 112)(6 38 59 74 95 17 97)(7 39 60 75 96 18 98)(8 40 61 76 81 19 99)(9 41 62 77 82 20 100)(10 42 63 78 83 21 101)(11 43 64 79 84 22 102)(12 44 49 80 85 23 103)(13 45 50 65 86 24 104)(14 46 51 66 87 25 105)(15 47 52 67 88 26 106)(16 48 53 68 89 27 107)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 22)(18 21)(19 20)(23 32)(24 31)(25 30)(26 29)(27 28)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(49 58)(50 57)(51 56)(52 55)(53 54)(59 64)(60 63)(61 62)(65 72)(66 71)(67 70)(68 69)(73 80)(74 79)(75 78)(76 77)(81 82)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(97 102)(98 101)(99 100)(103 112)(104 111)(105 110)(106 109)(107 108)
G:=sub<Sym(112)| (1,33,54,69,90,28,108)(2,34,55,70,91,29,109)(3,35,56,71,92,30,110)(4,36,57,72,93,31,111)(5,37,58,73,94,32,112)(6,38,59,74,95,17,97)(7,39,60,75,96,18,98)(8,40,61,76,81,19,99)(9,41,62,77,82,20,100)(10,42,63,78,83,21,101)(11,43,64,79,84,22,102)(12,44,49,80,85,23,103)(13,45,50,65,86,24,104)(14,46,51,66,87,25,105)(15,47,52,67,88,26,106)(16,48,53,68,89,27,107), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,22)(18,21)(19,20)(23,32)(24,31)(25,30)(26,29)(27,28)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(49,58)(50,57)(51,56)(52,55)(53,54)(59,64)(60,63)(61,62)(65,72)(66,71)(67,70)(68,69)(73,80)(74,79)(75,78)(76,77)(81,82)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(97,102)(98,101)(99,100)(103,112)(104,111)(105,110)(106,109)(107,108)>;
G:=Group( (1,33,54,69,90,28,108)(2,34,55,70,91,29,109)(3,35,56,71,92,30,110)(4,36,57,72,93,31,111)(5,37,58,73,94,32,112)(6,38,59,74,95,17,97)(7,39,60,75,96,18,98)(8,40,61,76,81,19,99)(9,41,62,77,82,20,100)(10,42,63,78,83,21,101)(11,43,64,79,84,22,102)(12,44,49,80,85,23,103)(13,45,50,65,86,24,104)(14,46,51,66,87,25,105)(15,47,52,67,88,26,106)(16,48,53,68,89,27,107), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,22)(18,21)(19,20)(23,32)(24,31)(25,30)(26,29)(27,28)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(49,58)(50,57)(51,56)(52,55)(53,54)(59,64)(60,63)(61,62)(65,72)(66,71)(67,70)(68,69)(73,80)(74,79)(75,78)(76,77)(81,82)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(97,102)(98,101)(99,100)(103,112)(104,111)(105,110)(106,109)(107,108) );
G=PermutationGroup([[(1,33,54,69,90,28,108),(2,34,55,70,91,29,109),(3,35,56,71,92,30,110),(4,36,57,72,93,31,111),(5,37,58,73,94,32,112),(6,38,59,74,95,17,97),(7,39,60,75,96,18,98),(8,40,61,76,81,19,99),(9,41,62,77,82,20,100),(10,42,63,78,83,21,101),(11,43,64,79,84,22,102),(12,44,49,80,85,23,103),(13,45,50,65,86,24,104),(14,46,51,66,87,25,105),(15,47,52,67,88,26,106),(16,48,53,68,89,27,107)], [(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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,22),(18,21),(19,20),(23,32),(24,31),(25,30),(26,29),(27,28),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(49,58),(50,57),(51,56),(52,55),(53,54),(59,64),(60,63),(61,62),(65,72),(66,71),(67,70),(68,69),(73,80),(74,79),(75,78),(76,77),(81,82),(83,96),(84,95),(85,94),(86,93),(87,92),(88,91),(89,90),(97,102),(98,101),(99,100),(103,112),(104,111),(105,110),(106,109),(107,108)]])
C7xD16 is a maximal subgroup of
C7:D32 D16.D7 D8:D14 D16:3D7
77 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 14G | ··· | 14R | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 56A | ··· | 56L | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 2 | 4 | 7 | ··· | 7 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 8 | 8 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | D4 | D8 | D16 | C7xD4 | C7xD8 | C7xD16 |
| kernel | C7xD16 | C112 | C7xD8 | D16 | C16 | D8 | C28 | C14 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 1 | 2 | 4 | 6 | 12 | 24 |
Matrix representation of C7xD16 ►in GL4(F113) generated by
| 106 | 0 | 0 | 0 |
| 0 | 106 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 82 | 0 | 0 |
| 62 | 51 | 0 | 0 |
| 0 | 0 | 18 | 109 |
| 0 | 0 | 4 | 18 |
| 51 | 31 | 0 | 0 |
| 51 | 62 | 0 | 0 |
| 0 | 0 | 18 | 109 |
| 0 | 0 | 109 | 95 |
G:=sub<GL(4,GF(113))| [106,0,0,0,0,106,0,0,0,0,1,0,0,0,0,1],[0,62,0,0,82,51,0,0,0,0,18,4,0,0,109,18],[51,51,0,0,31,62,0,0,0,0,18,109,0,0,109,95] >;
C7xD16 in GAP, Magma, Sage, TeX
C_7\times D_{16} % in TeX
G:=Group("C7xD16"); // GroupNames label
G:=SmallGroup(224,60);
// by ID
G=gap.SmallGroup(224,60);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-2,361,2019,1017,165,5044,2530,88]);
// Polycyclic
G:=Group<a,b,c|a^7=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export