metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C113⋊C4, D113.C2, SmallGroup(452,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C113 — C113⋊C4 |
Generators and relations for C113⋊C4
G = < a,b | a113=b4=1, bab-1=a15 >
Smallest permutation representation of C113⋊C4
►On 113 points: primitive
Generators in S113
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113)
(2 99 113 16)(3 84 112 31)(4 69 111 46)(5 54 110 61)(6 39 109 76)(7 24 108 91)(8 9 107 106)(10 92 105 23)(11 77 104 38)(12 62 103 53)(13 47 102 68)(14 32 101 83)(15 17 100 98)(18 85 97 30)(19 70 96 45)(20 55 95 60)(21 40 94 75)(22 25 93 90)(26 78 89 37)(27 63 88 52)(28 48 87 67)(29 33 86 82)(34 71 81 44)(35 56 80 59)(36 41 79 74)(42 64 73 51)(43 49 72 66)(50 57 65 58)
G:=sub<Sym(113)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113), (2,99,113,16)(3,84,112,31)(4,69,111,46)(5,54,110,61)(6,39,109,76)(7,24,108,91)(8,9,107,106)(10,92,105,23)(11,77,104,38)(12,62,103,53)(13,47,102,68)(14,32,101,83)(15,17,100,98)(18,85,97,30)(19,70,96,45)(20,55,95,60)(21,40,94,75)(22,25,93,90)(26,78,89,37)(27,63,88,52)(28,48,87,67)(29,33,86,82)(34,71,81,44)(35,56,80,59)(36,41,79,74)(42,64,73,51)(43,49,72,66)(50,57,65,58)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113), (2,99,113,16)(3,84,112,31)(4,69,111,46)(5,54,110,61)(6,39,109,76)(7,24,108,91)(8,9,107,106)(10,92,105,23)(11,77,104,38)(12,62,103,53)(13,47,102,68)(14,32,101,83)(15,17,100,98)(18,85,97,30)(19,70,96,45)(20,55,95,60)(21,40,94,75)(22,25,93,90)(26,78,89,37)(27,63,88,52)(28,48,87,67)(29,33,86,82)(34,71,81,44)(35,56,80,59)(36,41,79,74)(42,64,73,51)(43,49,72,66)(50,57,65,58) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113)], [(2,99,113,16),(3,84,112,31),(4,69,111,46),(5,54,110,61),(6,39,109,76),(7,24,108,91),(8,9,107,106),(10,92,105,23),(11,77,104,38),(12,62,103,53),(13,47,102,68),(14,32,101,83),(15,17,100,98),(18,85,97,30),(19,70,96,45),(20,55,95,60),(21,40,94,75),(22,25,93,90),(26,78,89,37),(27,63,88,52),(28,48,87,67),(29,33,86,82),(34,71,81,44),(35,56,80,59),(36,41,79,74),(42,64,73,51),(43,49,72,66),(50,57,65,58)]])
32 conjugacy classes
| class | 1 | 2 | 4A | 4B | 113A | ··· | 113AB |
| order | 1 | 2 | 4 | 4 | 113 | ··· | 113 |
| size | 1 | 113 | 113 | 113 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 4 |
| type | + | + | + | |
| image | C1 | C2 | C4 | C113⋊C4 |
| kernel | C113⋊C4 | D113 | C113 | C1 |
| # reps | 1 | 1 | 2 | 28 |
Matrix representation of C113⋊C4 ►in GL4(𝔽2713) generated by
| 1866 | 1 | 0 | 0 |
| 2495 | 0 | 1 | 0 |
| 291 | 0 | 0 | 1 |
| 2247 | 1476 | 1245 | 892 |
| 283 | 1765 | 449 | 1870 |
| 1678 | 700 | 2407 | 1568 |
| 20 | 791 | 1755 | 1402 |
| 2364 | 2575 | 223 | 2688 |
G:=sub<GL(4,GF(2713))| [1866,2495,291,2247,1,0,0,1476,0,1,0,1245,0,0,1,892],[283,1678,20,2364,1765,700,791,2575,449,2407,1755,223,1870,1568,1402,2688] >;
C113⋊C4 in GAP, Magma, Sage, TeX
C_{113}\rtimes C_4 % in TeX
G:=Group("C113:C4"); // GroupNames label
G:=SmallGroup(452,3);
// by ID
G=gap.SmallGroup(452,3);
# by ID
G:=PCGroup([3,-2,-2,-113,6,3530,2021]);
// Polycyclic
G:=Group<a,b|a^113=b^4=1,b*a*b^-1=a^15>;
// generators/relations
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