metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C112⋊1C4, C56.1Q8, C16⋊1Dic7, C28.6SD16, C8.3Dic14, M5(2).1D7, C7⋊3(C8.Q8), (C2×C4).8D28, C56.72(C2×C4), (C2×C28).98D4, (C2×C8).45D14, C28.26(C4⋊C4), C8⋊Dic7.1C2, C8.18(C2×Dic7), (C2×C14).7SD16, C14.6(C4.Q8), C2.3(C8⋊Dic7), C56.C4.5C2, C4.11(C56⋊C2), (C2×C56).49C22, C4.11(C4⋊Dic7), (C7×M5(2)).1C2, C22.5(C56⋊C2), SmallGroup(448,70)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C16⋊Dic7
G = < a,b,c | a16=b14=1, c2=b7, bab-1=a9, cac-1=a11, cbc-1=b-1 >
Smallest permutation representation of C16⋊Dic7
►On 112 points
Generators in S112
(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 93 78 49 100 21 47)(2 86 79 58 101 30 48 10 94 71 50 109 22 40)(3 95 80 51 102 23 33)(4 88 65 60 103 32 34 12 96 73 52 111 24 42)(5 81 66 53 104 25 35)(6 90 67 62 105 18 36 14 82 75 54 97 26 44)(7 83 68 55 106 27 37)(8 92 69 64 107 20 38 16 84 77 56 99 28 46)(9 85 70 57 108 29 39)(11 87 72 59 110 31 41)(13 89 74 61 112 17 43)(15 91 76 63 98 19 45)
(2 4 10 12)(3 7)(5 13)(6 16 14 8)(11 15)(17 66)(18 69 26 77)(19 72)(20 75 28 67)(21 78)(22 65 30 73)(23 68)(24 71 32 79)(25 74)(27 80)(29 70)(31 76)(33 83)(34 86 42 94)(35 89)(36 92 44 84)(37 95)(38 82 46 90)(39 85)(40 88 48 96)(41 91)(43 81)(45 87)(47 93)(49 100)(50 103 58 111)(51 106)(52 109 60 101)(53 112)(54 99 62 107)(55 102)(56 105 64 97)(57 108)(59 98)(61 104)(63 110)
G:=sub<Sym(112)| (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), (1,93,78,49,100,21,47)(2,86,79,58,101,30,48,10,94,71,50,109,22,40)(3,95,80,51,102,23,33)(4,88,65,60,103,32,34,12,96,73,52,111,24,42)(5,81,66,53,104,25,35)(6,90,67,62,105,18,36,14,82,75,54,97,26,44)(7,83,68,55,106,27,37)(8,92,69,64,107,20,38,16,84,77,56,99,28,46)(9,85,70,57,108,29,39)(11,87,72,59,110,31,41)(13,89,74,61,112,17,43)(15,91,76,63,98,19,45), (2,4,10,12)(3,7)(5,13)(6,16,14,8)(11,15)(17,66)(18,69,26,77)(19,72)(20,75,28,67)(21,78)(22,65,30,73)(23,68)(24,71,32,79)(25,74)(27,80)(29,70)(31,76)(33,83)(34,86,42,94)(35,89)(36,92,44,84)(37,95)(38,82,46,90)(39,85)(40,88,48,96)(41,91)(43,81)(45,87)(47,93)(49,100)(50,103,58,111)(51,106)(52,109,60,101)(53,112)(54,99,62,107)(55,102)(56,105,64,97)(57,108)(59,98)(61,104)(63,110)>;
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), (1,93,78,49,100,21,47)(2,86,79,58,101,30,48,10,94,71,50,109,22,40)(3,95,80,51,102,23,33)(4,88,65,60,103,32,34,12,96,73,52,111,24,42)(5,81,66,53,104,25,35)(6,90,67,62,105,18,36,14,82,75,54,97,26,44)(7,83,68,55,106,27,37)(8,92,69,64,107,20,38,16,84,77,56,99,28,46)(9,85,70,57,108,29,39)(11,87,72,59,110,31,41)(13,89,74,61,112,17,43)(15,91,76,63,98,19,45), (2,4,10,12)(3,7)(5,13)(6,16,14,8)(11,15)(17,66)(18,69,26,77)(19,72)(20,75,28,67)(21,78)(22,65,30,73)(23,68)(24,71,32,79)(25,74)(27,80)(29,70)(31,76)(33,83)(34,86,42,94)(35,89)(36,92,44,84)(37,95)(38,82,46,90)(39,85)(40,88,48,96)(41,91)(43,81)(45,87)(47,93)(49,100)(50,103,58,111)(51,106)(52,109,60,101)(53,112)(54,99,62,107)(55,102)(56,105,64,97)(57,108)(59,98)(61,104)(63,110) );
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)], [(1,93,78,49,100,21,47),(2,86,79,58,101,30,48,10,94,71,50,109,22,40),(3,95,80,51,102,23,33),(4,88,65,60,103,32,34,12,96,73,52,111,24,42),(5,81,66,53,104,25,35),(6,90,67,62,105,18,36,14,82,75,54,97,26,44),(7,83,68,55,106,27,37),(8,92,69,64,107,20,38,16,84,77,56,99,28,46),(9,85,70,57,108,29,39),(11,87,72,59,110,31,41),(13,89,74,61,112,17,43),(15,91,76,63,98,19,45)], [(2,4,10,12),(3,7),(5,13),(6,16,14,8),(11,15),(17,66),(18,69,26,77),(19,72),(20,75,28,67),(21,78),(22,65,30,73),(23,68),(24,71,32,79),(25,74),(27,80),(29,70),(31,76),(33,83),(34,86,42,94),(35,89),(36,92,44,84),(37,95),(38,82,46,90),(39,85),(40,88,48,96),(41,91),(43,81),(45,87),(47,93),(49,100),(50,103,58,111),(51,106),(52,109,60,101),(53,112),(54,99,62,107),(55,102),(56,105,64,97),(57,108),(59,98),(61,104),(63,110)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | 14E | 14F | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56R | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 2 | 2 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 4 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | + | - | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | D7 | SD16 | SD16 | Dic7 | D14 | Dic14 | D28 | C56⋊C2 | C56⋊C2 | C8.Q8 | C16⋊Dic7 |
| kernel | C16⋊Dic7 | C8⋊Dic7 | C56.C4 | C7×M5(2) | C112 | C56 | C2×C28 | M5(2) | C28 | C2×C14 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 6 | 3 | 6 | 6 | 12 | 12 | 2 | 12 |
Matrix representation of C16⋊Dic7 ►in GL4(𝔽113) generated by
| 50 | 42 | 111 | 0 |
| 71 | 89 | 0 | 111 |
| 4 | 73 | 63 | 71 |
| 40 | 96 | 42 | 24 |
| 0 | 1 | 0 | 0 |
| 112 | 9 | 0 | 0 |
| 71 | 89 | 0 | 112 |
| 24 | 81 | 1 | 104 |
| 1 | 0 | 0 | 0 |
| 9 | 112 | 0 | 0 |
| 17 | 73 | 63 | 71 |
| 0 | 96 | 73 | 50 |
G:=sub<GL(4,GF(113))| [50,71,4,40,42,89,73,96,111,0,63,42,0,111,71,24],[0,112,71,24,1,9,89,81,0,0,0,1,0,0,112,104],[1,9,17,0,0,112,73,96,0,0,63,73,0,0,71,50] >;
C16⋊Dic7 in GAP, Magma, Sage, TeX
C_{16}\rtimes {\rm Dic}_7 % in TeX
G:=Group("C16:Dic7"); // GroupNames label
G:=SmallGroup(448,70);
// by ID
G=gap.SmallGroup(448,70);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,64,387,675,80,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^16=b^14=1,c^2=b^7,b*a*b^-1=a^9,c*a*c^-1=a^11,c*b*c^-1=b^-1>;
// generators/relations
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