metacyclic, supersoluble, monomial, Z-group
Aliases: C7⋊C48, C14.C24, C8.2F7, C56.2C6, C28.2C12, C7⋊C3⋊C16, C7⋊C16⋊C3, C2.(C7⋊C24), C4.2(C7⋊C12), (C2×C7⋊C3).C8, (C4×C7⋊C3).2C4, (C8×C7⋊C3).2C2, SmallGroup(336,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7⋊C48 |
Generators and relations for C7⋊C48
G = < a,b | a7=b48=1, bab-1=a5 >
Smallest permutation representation of C7⋊C48
►On 112 points
Generators in S112
(1 41 25 103 57 71 87)(2 72 104 42 88 58 26)(3 59 43 73 27 89 105)(4 90 74 60 106 28 44)(5 29 61 91 45 107 75)(6 108 92 30 76 46 62)(7 47 31 109 63 77 93)(8 78 110 48 94 64 32)(9 17 49 79 33 95 111)(10 96 80 18 112 34 50)(11 35 19 97 51 65 81)(12 66 98 36 82 52 20)(13 53 37 67 21 83 99)(14 84 68 54 100 22 38)(15 23 55 85 39 101 69)(16 102 86 24 70 40 56)
(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)| (1,41,25,103,57,71,87)(2,72,104,42,88,58,26)(3,59,43,73,27,89,105)(4,90,74,60,106,28,44)(5,29,61,91,45,107,75)(6,108,92,30,76,46,62)(7,47,31,109,63,77,93)(8,78,110,48,94,64,32)(9,17,49,79,33,95,111)(10,96,80,18,112,34,50)(11,35,19,97,51,65,81)(12,66,98,36,82,52,20)(13,53,37,67,21,83,99)(14,84,68,54,100,22,38)(15,23,55,85,39,101,69)(16,102,86,24,70,40,56), (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( (1,41,25,103,57,71,87)(2,72,104,42,88,58,26)(3,59,43,73,27,89,105)(4,90,74,60,106,28,44)(5,29,61,91,45,107,75)(6,108,92,30,76,46,62)(7,47,31,109,63,77,93)(8,78,110,48,94,64,32)(9,17,49,79,33,95,111)(10,96,80,18,112,34,50)(11,35,19,97,51,65,81)(12,66,98,36,82,52,20)(13,53,37,67,21,83,99)(14,84,68,54,100,22,38)(15,23,55,85,39,101,69)(16,102,86,24,70,40,56), (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([[(1,41,25,103,57,71,87),(2,72,104,42,88,58,26),(3,59,43,73,27,89,105),(4,90,74,60,106,28,44),(5,29,61,91,45,107,75),(6,108,92,30,76,46,62),(7,47,31,109,63,77,93),(8,78,110,48,94,64,32),(9,17,49,79,33,95,111),(10,96,80,18,112,34,50),(11,35,19,97,51,65,81),(12,66,98,36,82,52,20),(13,53,37,67,21,83,99),(14,84,68,54,100,22,38),(15,23,55,85,39,101,69),(16,102,86,24,70,40,56)], [(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)]])
56 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 7 | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 14 | 16A | ··· | 16H | 24A | ··· | 24H | 28A | 28B | 48A | ··· | 48P | 56A | 56B | 56C | 56D |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 14 | 16 | ··· | 16 | 24 | ··· | 24 | 28 | 28 | 48 | ··· | 48 | 56 | 56 | 56 | 56 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 7 | 7 | 6 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 6 | 7 | ··· | 7 | 7 | ··· | 7 | 6 | 6 | 7 | ··· | 7 | 6 | 6 | 6 | 6 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 |
| type | + | + | + | - | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C48 | F7 | C7⋊C12 | C7⋊C24 | C7⋊C48 |
| kernel | C7⋊C48 | C8×C7⋊C3 | C7⋊C16 | C4×C7⋊C3 | C56 | C2×C7⋊C3 | C28 | C7⋊C3 | C14 | C7 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 1 | 1 | 2 | 4 |
Matrix representation of C7⋊C48 ►in GL6(𝔽337)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 336 | 336 | 336 | 336 | 336 | 336 |
| 227 | 0 | 227 | 227 | 234 | 0 |
| 227 | 227 | 234 | 0 | 0 | 227 |
| 7 | 110 | 110 | 0 | 110 | 0 |
| 0 | 227 | 0 | 227 | 227 | 234 |
| 110 | 0 | 0 | 7 | 110 | 110 |
| 0 | 7 | 110 | 110 | 0 | 110 |
G:=sub<GL(6,GF(337))| [0,0,0,0,0,336,1,0,0,0,0,336,0,1,0,0,0,336,0,0,1,0,0,336,0,0,0,1,0,336,0,0,0,0,1,336],[227,227,7,0,110,0,0,227,110,227,0,7,227,234,110,0,0,110,227,0,0,227,7,110,234,0,110,227,110,0,0,227,0,234,110,110] >;
C7⋊C48 in GAP, Magma, Sage, TeX
C_7\rtimes C_{48} % in TeX
G:=Group("C7:C48"); // GroupNames label
G:=SmallGroup(336,1);
// by ID
G=gap.SmallGroup(336,1);
# by ID
G:=PCGroup([6,-2,-3,-2,-2,-2,-7,36,50,69,10373,3467]);
// Polycyclic
G:=Group<a,b|a^7=b^48=1,b*a*b^-1=a^5>;
// generators/relations
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