metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7⋊C16, C14.C8, C8.2D7, C28.2C4, C56.2C2, C4.2Dic7, C2.(C7⋊C8), SmallGroup(112,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7⋊C16 |
Generators and relations for C7⋊C16
G = < a,b | a7=b16=1, bab-1=a-1 >
Smallest permutation representation of C7⋊C16
►Regular action on 112 points
Generators in S112
(1 42 102 88 73 24 53)(2 54 25 74 89 103 43)(3 44 104 90 75 26 55)(4 56 27 76 91 105 45)(5 46 106 92 77 28 57)(6 58 29 78 93 107 47)(7 48 108 94 79 30 59)(8 60 31 80 95 109 33)(9 34 110 96 65 32 61)(10 62 17 66 81 111 35)(11 36 112 82 67 18 63)(12 64 19 68 83 97 37)(13 38 98 84 69 20 49)(14 50 21 70 85 99 39)(15 40 100 86 71 22 51)(16 52 23 72 87 101 41)
(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)
G:=sub<Sym(112)| (1,42,102,88,73,24,53)(2,54,25,74,89,103,43)(3,44,104,90,75,26,55)(4,56,27,76,91,105,45)(5,46,106,92,77,28,57)(6,58,29,78,93,107,47)(7,48,108,94,79,30,59)(8,60,31,80,95,109,33)(9,34,110,96,65,32,61)(10,62,17,66,81,111,35)(11,36,112,82,67,18,63)(12,64,19,68,83,97,37)(13,38,98,84,69,20,49)(14,50,21,70,85,99,39)(15,40,100,86,71,22,51)(16,52,23,72,87,101,41), (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)>;
G:=Group( (1,42,102,88,73,24,53)(2,54,25,74,89,103,43)(3,44,104,90,75,26,55)(4,56,27,76,91,105,45)(5,46,106,92,77,28,57)(6,58,29,78,93,107,47)(7,48,108,94,79,30,59)(8,60,31,80,95,109,33)(9,34,110,96,65,32,61)(10,62,17,66,81,111,35)(11,36,112,82,67,18,63)(12,64,19,68,83,97,37)(13,38,98,84,69,20,49)(14,50,21,70,85,99,39)(15,40,100,86,71,22,51)(16,52,23,72,87,101,41), (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) );
G=PermutationGroup([[(1,42,102,88,73,24,53),(2,54,25,74,89,103,43),(3,44,104,90,75,26,55),(4,56,27,76,91,105,45),(5,46,106,92,77,28,57),(6,58,29,78,93,107,47),(7,48,108,94,79,30,59),(8,60,31,80,95,109,33),(9,34,110,96,65,32,61),(10,62,17,66,81,111,35),(11,36,112,82,67,18,63),(12,64,19,68,83,97,37),(13,38,98,84,69,20,49),(14,50,21,70,85,99,39),(15,40,100,86,71,22,51),(16,52,23,72,87,101,41)], [(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)]])
C7⋊C16 is a maximal subgroup of
D7×C16 C16⋊D7 C28.C8 C7⋊D16 D8.D7 C7⋊SD32 C7⋊Q32 C7⋊C48 C21⋊C16
C7⋊C16 is a maximal quotient of
C7⋊C32 C21⋊C16
40 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 16A | ··· | 16H | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||
| image | C1 | C2 | C4 | C8 | C16 | D7 | Dic7 | C7⋊C8 | C7⋊C16 |
| kernel | C7⋊C16 | C56 | C28 | C14 | C7 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 3 | 3 | 6 | 12 |
Matrix representation of C7⋊C16 ►in GL2(𝔽41) generated by
| 40 | 40 |
| 32 | 31 |
| 35 | 6 |
| 15 | 6 |
G:=sub<GL(2,GF(41))| [40,32,40,31],[35,15,6,6] >;
C7⋊C16 in GAP, Magma, Sage, TeX
C_7\rtimes C_{16} % in TeX
G:=Group("C7:C16"); // GroupNames label
G:=SmallGroup(112,1);
// by ID
G=gap.SmallGroup(112,1);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,10,26,42,2404]);
// Polycyclic
G:=Group<a,b|a^7=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
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