metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52.56D4, Q8⋊2Dic13, D4⋊2Dic13, C13⋊5C4≀C2, (D4×C13)⋊5C4, (Q8×C13)⋊5C4, (C2×C26).3D4, C52.30(C2×C4), C4○D4.1D13, (C2×C4).41D26, C52.4C4⋊4C2, (C4×Dic13)⋊2C2, C4.3(C2×Dic13), C4.31(C13⋊D4), (C2×C52).20C22, C26.29(C22⋊C4), C22.3(C13⋊D4), C2.8(C23.D13), (C13×C4○D4).1C2, SmallGroup(416,44)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.56D4
G = < a,b,c | a52=b4=1, c2=a13, bab-1=cac-1=a25, cbc-1=a13b-1 >
Smallest permutation representation of C52.56D4
►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 14 27 40)(2 39 28 13)(3 12 29 38)(4 37 30 11)(5 10 31 36)(6 35 32 9)(7 8 33 34)(15 52 41 26)(16 25 42 51)(17 50 43 24)(18 23 44 49)(19 48 45 22)(20 21 46 47)(53 89)(54 62)(55 87)(56 60)(57 85)(59 83)(61 81)(63 79)(64 104)(65 77)(66 102)(67 75)(68 100)(69 73)(70 98)(72 96)(74 94)(76 92)(78 90)(80 88)(82 86)(91 103)(93 101)(95 99)
(1 58 14 71 27 84 40 97)(2 83 15 96 28 57 41 70)(3 56 16 69 29 82 42 95)(4 81 17 94 30 55 43 68)(5 54 18 67 31 80 44 93)(6 79 19 92 32 53 45 66)(7 104 20 65 33 78 46 91)(8 77 21 90 34 103 47 64)(9 102 22 63 35 76 48 89)(10 75 23 88 36 101 49 62)(11 100 24 61 37 74 50 87)(12 73 25 86 38 99 51 60)(13 98 26 59 39 72 52 85)
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), (1,14,27,40)(2,39,28,13)(3,12,29,38)(4,37,30,11)(5,10,31,36)(6,35,32,9)(7,8,33,34)(15,52,41,26)(16,25,42,51)(17,50,43,24)(18,23,44,49)(19,48,45,22)(20,21,46,47)(53,89)(54,62)(55,87)(56,60)(57,85)(59,83)(61,81)(63,79)(64,104)(65,77)(66,102)(67,75)(68,100)(69,73)(70,98)(72,96)(74,94)(76,92)(78,90)(80,88)(82,86)(91,103)(93,101)(95,99), (1,58,14,71,27,84,40,97)(2,83,15,96,28,57,41,70)(3,56,16,69,29,82,42,95)(4,81,17,94,30,55,43,68)(5,54,18,67,31,80,44,93)(6,79,19,92,32,53,45,66)(7,104,20,65,33,78,46,91)(8,77,21,90,34,103,47,64)(9,102,22,63,35,76,48,89)(10,75,23,88,36,101,49,62)(11,100,24,61,37,74,50,87)(12,73,25,86,38,99,51,60)(13,98,26,59,39,72,52,85)>;
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), (1,14,27,40)(2,39,28,13)(3,12,29,38)(4,37,30,11)(5,10,31,36)(6,35,32,9)(7,8,33,34)(15,52,41,26)(16,25,42,51)(17,50,43,24)(18,23,44,49)(19,48,45,22)(20,21,46,47)(53,89)(54,62)(55,87)(56,60)(57,85)(59,83)(61,81)(63,79)(64,104)(65,77)(66,102)(67,75)(68,100)(69,73)(70,98)(72,96)(74,94)(76,92)(78,90)(80,88)(82,86)(91,103)(93,101)(95,99), (1,58,14,71,27,84,40,97)(2,83,15,96,28,57,41,70)(3,56,16,69,29,82,42,95)(4,81,17,94,30,55,43,68)(5,54,18,67,31,80,44,93)(6,79,19,92,32,53,45,66)(7,104,20,65,33,78,46,91)(8,77,21,90,34,103,47,64)(9,102,22,63,35,76,48,89)(10,75,23,88,36,101,49,62)(11,100,24,61,37,74,50,87)(12,73,25,86,38,99,51,60)(13,98,26,59,39,72,52,85) );
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)], [(1,14,27,40),(2,39,28,13),(3,12,29,38),(4,37,30,11),(5,10,31,36),(6,35,32,9),(7,8,33,34),(15,52,41,26),(16,25,42,51),(17,50,43,24),(18,23,44,49),(19,48,45,22),(20,21,46,47),(53,89),(54,62),(55,87),(56,60),(57,85),(59,83),(61,81),(63,79),(64,104),(65,77),(66,102),(67,75),(68,100),(69,73),(70,98),(72,96),(74,94),(76,92),(78,90),(80,88),(82,86),(91,103),(93,101),(95,99)], [(1,58,14,71,27,84,40,97),(2,83,15,96,28,57,41,70),(3,56,16,69,29,82,42,95),(4,81,17,94,30,55,43,68),(5,54,18,67,31,80,44,93),(6,79,19,92,32,53,45,66),(7,104,20,65,33,78,46,91),(8,77,21,90,34,103,47,64),(9,102,22,63,35,76,48,89),(10,75,23,88,36,101,49,62),(11,100,24,61,37,74,50,87),(12,73,25,86,38,99,51,60),(13,98,26,59,39,72,52,85)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26X | 52A | ··· | 52L | 52M | ··· | 52AD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 26 | 26 | 26 | 26 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D13 | C4≀C2 | D26 | Dic13 | Dic13 | C13⋊D4 | C13⋊D4 | C52.56D4 |
| kernel | C52.56D4 | C52.4C4 | C4×Dic13 | C13×C4○D4 | D4×C13 | Q8×C13 | C52 | C2×C26 | C4○D4 | C13 | C2×C4 | D4 | Q8 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 6 | 4 | 6 | 6 | 6 | 12 | 12 | 12 |
Matrix representation of C52.56D4 ►in GL4(𝔽313) generated by
| 192 | 78 | 0 | 0 |
| 2 | 97 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 65 | 222 | 0 | 0 |
| 184 | 248 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 80 | 1 |
| 99 | 215 | 0 | 0 |
| 215 | 214 | 0 | 0 |
| 0 | 0 | 68 | 230 |
| 0 | 0 | 161 | 245 |
G:=sub<GL(4,GF(313))| [192,2,0,0,78,97,0,0,0,0,25,0,0,0,0,25],[65,184,0,0,222,248,0,0,0,0,25,80,0,0,0,1],[99,215,0,0,215,214,0,0,0,0,68,161,0,0,230,245] >;
C52.56D4 in GAP, Magma, Sage, TeX
C_{52}._{56}D_4 % in TeX
G:=Group("C52.56D4"); // GroupNames label
G:=SmallGroup(416,44);
// by ID
G=gap.SmallGroup(416,44);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,86,579,297,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=b^4=1,c^2=a^13,b*a*b^-1=c*a*c^-1=a^25,c*b*c^-1=a^13*b^-1>;
// generators/relations
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