metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52.8D4, C23.Dic13, (C2×C4).3D26, (C2×D4).2D13, (D4×C26).2C2, C52.4C4⋊3C2, C13⋊3(C4.D4), (C22×C26).2C4, C4.13(C13⋊D4), (C2×C52).17C22, C26.25(C22⋊C4), C2.4(C23.D13), C22.2(C2×Dic13), (C2×C26).48(C2×C4), SmallGroup(416,40)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.D4
G = < a,b,c | a52=1, b4=a26, c2=a13, bab-1=a-1, cac-1=a25, cbc-1=a39b3 >
Smallest permutation representation of C52.D4
►On 104 points
Generators in S104
(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)
(1 68 14 55 27 94 40 81)(2 67 15 54 28 93 41 80)(3 66 16 53 29 92 42 79)(4 65 17 104 30 91 43 78)(5 64 18 103 31 90 44 77)(6 63 19 102 32 89 45 76)(7 62 20 101 33 88 46 75)(8 61 21 100 34 87 47 74)(9 60 22 99 35 86 48 73)(10 59 23 98 36 85 49 72)(11 58 24 97 37 84 50 71)(12 57 25 96 38 83 51 70)(13 56 26 95 39 82 52 69)
(1 94 14 55 27 68 40 81)(2 67 15 80 28 93 41 54)(3 92 16 53 29 66 42 79)(4 65 17 78 30 91 43 104)(5 90 18 103 31 64 44 77)(6 63 19 76 32 89 45 102)(7 88 20 101 33 62 46 75)(8 61 21 74 34 87 47 100)(9 86 22 99 35 60 48 73)(10 59 23 72 36 85 49 98)(11 84 24 97 37 58 50 71)(12 57 25 70 38 83 51 96)(13 82 26 95 39 56 52 69)
G:=sub<Sym(104)| (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), (1,68,14,55,27,94,40,81)(2,67,15,54,28,93,41,80)(3,66,16,53,29,92,42,79)(4,65,17,104,30,91,43,78)(5,64,18,103,31,90,44,77)(6,63,19,102,32,89,45,76)(7,62,20,101,33,88,46,75)(8,61,21,100,34,87,47,74)(9,60,22,99,35,86,48,73)(10,59,23,98,36,85,49,72)(11,58,24,97,37,84,50,71)(12,57,25,96,38,83,51,70)(13,56,26,95,39,82,52,69), (1,94,14,55,27,68,40,81)(2,67,15,80,28,93,41,54)(3,92,16,53,29,66,42,79)(4,65,17,78,30,91,43,104)(5,90,18,103,31,64,44,77)(6,63,19,76,32,89,45,102)(7,88,20,101,33,62,46,75)(8,61,21,74,34,87,47,100)(9,86,22,99,35,60,48,73)(10,59,23,72,36,85,49,98)(11,84,24,97,37,58,50,71)(12,57,25,70,38,83,51,96)(13,82,26,95,39,56,52,69)>;
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), (1,68,14,55,27,94,40,81)(2,67,15,54,28,93,41,80)(3,66,16,53,29,92,42,79)(4,65,17,104,30,91,43,78)(5,64,18,103,31,90,44,77)(6,63,19,102,32,89,45,76)(7,62,20,101,33,88,46,75)(8,61,21,100,34,87,47,74)(9,60,22,99,35,86,48,73)(10,59,23,98,36,85,49,72)(11,58,24,97,37,84,50,71)(12,57,25,96,38,83,51,70)(13,56,26,95,39,82,52,69), (1,94,14,55,27,68,40,81)(2,67,15,80,28,93,41,54)(3,92,16,53,29,66,42,79)(4,65,17,78,30,91,43,104)(5,90,18,103,31,64,44,77)(6,63,19,76,32,89,45,102)(7,88,20,101,33,62,46,75)(8,61,21,74,34,87,47,100)(9,86,22,99,35,60,48,73)(10,59,23,72,36,85,49,98)(11,84,24,97,37,58,50,71)(12,57,25,70,38,83,51,96)(13,82,26,95,39,56,52,69) );
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)], [(1,68,14,55,27,94,40,81),(2,67,15,54,28,93,41,80),(3,66,16,53,29,92,42,79),(4,65,17,104,30,91,43,78),(5,64,18,103,31,90,44,77),(6,63,19,102,32,89,45,76),(7,62,20,101,33,88,46,75),(8,61,21,100,34,87,47,74),(9,60,22,99,35,86,48,73),(10,59,23,98,36,85,49,72),(11,58,24,97,37,84,50,71),(12,57,25,96,38,83,51,70),(13,56,26,95,39,82,52,69)], [(1,94,14,55,27,68,40,81),(2,67,15,80,28,93,41,54),(3,92,16,53,29,66,42,79),(4,65,17,78,30,91,43,104),(5,90,18,103,31,64,44,77),(6,63,19,76,32,89,45,102),(7,88,20,101,33,62,46,75),(8,61,21,74,34,87,47,100),(9,86,22,99,35,60,48,73),(10,59,23,72,36,85,49,98),(11,84,24,97,37,58,50,71),(12,57,25,70,38,83,51,96),(13,82,26,95,39,56,52,69)]])
71 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26R | 26S | ··· | 26AP | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 52 | 52 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
71 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | D4 | D13 | D26 | Dic13 | C13⋊D4 | C4.D4 | C52.D4 |
| kernel | C52.D4 | C52.4C4 | D4×C26 | C22×C26 | C52 | C2×D4 | C2×C4 | C23 | C4 | C13 | C1 |
| # reps | 1 | 2 | 1 | 4 | 2 | 6 | 6 | 12 | 24 | 1 | 12 |
Matrix representation of C52.D4 ►in GL4(𝔽313) generated by
| 4 | 57 | 50 | 120 |
| 235 | 309 | 93 | 101 |
| 0 | 0 | 0 | 280 |
| 0 | 0 | 33 | 0 |
| 110 | 59 | 306 | 291 |
| 0 | 0 | 0 | 312 |
| 0 | 1 | 0 | 0 |
| 243 | 181 | 252 | 203 |
| 203 | 136 | 159 | 22 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 70 | 132 | 61 | 110 |
G:=sub<GL(4,GF(313))| [4,235,0,0,57,309,0,0,50,93,0,33,120,101,280,0],[110,0,0,243,59,0,1,181,306,0,0,252,291,312,0,203],[203,0,0,70,136,0,1,132,159,0,0,61,22,1,0,110] >;
C52.D4 in GAP, Magma, Sage, TeX
C_{52}.D_4 % in TeX
G:=Group("C52.D4"); // GroupNames label
G:=SmallGroup(416,40);
// by ID
G=gap.SmallGroup(416,40);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,188,86,579,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=1,b^4=a^26,c^2=a^13,b*a*b^-1=a^-1,c*a*c^-1=a^25,c*b*c^-1=a^39*b^3>;
// generators/relations
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