metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C52.1C4, D26.3C4, C13⋊1M4(2), Dic13.5C22, C13⋊C8⋊1C2, C4.(C13⋊C4), C26.2(C2×C4), (C4×D13).3C2, C2.4(C2×C13⋊C4), SmallGroup(208,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.C4
G = < a,b | a52=1, b4=a26, bab-1=a31 >
Character table of C52.C4
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 13A | 13B | 13C | 26A | 26B | 26C | 52A | 52B | 52C | 52D | 52E | 52F | |
| size | 1 | 1 | 26 | 2 | 13 | 13 | 26 | 26 | 26 | 26 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | -1 | -i | i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | -1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | -1 | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | -1 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ10 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ11 | 4 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | orthogonal lifted from C2×C13⋊C4 |
| ρ12 | 4 | 4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | orthogonal lifted from C13⋊C4 |
| ρ13 | 4 | 4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | orthogonal lifted from C13⋊C4 |
| ρ14 | 4 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | orthogonal lifted from C2×C13⋊C4 |
| ρ15 | 4 | 4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | orthogonal lifted from C13⋊C4 |
| ρ16 | 4 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | orthogonal lifted from C2×C13⋊C4 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | ζ4ζ1312-ζ4ζ138-ζ4ζ135+ζ4ζ13 | -ζ4ζ1311+ζ4ζ1310+ζ4ζ133-ζ4ζ132 | ζ43ζ139-ζ43ζ137-ζ43ζ136+ζ43ζ134 | ζ4ζ1311-ζ4ζ1310-ζ4ζ133+ζ4ζ132 | -ζ43ζ139+ζ43ζ137+ζ43ζ136-ζ43ζ134 | ζ43ζ1312-ζ43ζ138-ζ43ζ135+ζ43ζ13 | complex faithful |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | ζ43ζ139-ζ43ζ137-ζ43ζ136+ζ43ζ134 | ζ43ζ1312-ζ43ζ138-ζ43ζ135+ζ43ζ13 | -ζ4ζ1311+ζ4ζ1310+ζ4ζ133-ζ4ζ132 | ζ4ζ1312-ζ4ζ138-ζ4ζ135+ζ4ζ13 | ζ4ζ1311-ζ4ζ1310-ζ4ζ133+ζ4ζ132 | -ζ43ζ139+ζ43ζ137+ζ43ζ136-ζ43ζ134 | complex faithful |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ43ζ139+ζ43ζ137+ζ43ζ136-ζ43ζ134 | ζ4ζ1312-ζ4ζ138-ζ4ζ135+ζ4ζ13 | ζ4ζ1311-ζ4ζ1310-ζ4ζ133+ζ4ζ132 | ζ43ζ1312-ζ43ζ138-ζ43ζ135+ζ43ζ13 | -ζ4ζ1311+ζ4ζ1310+ζ4ζ133-ζ4ζ132 | ζ43ζ139-ζ43ζ137-ζ43ζ136+ζ43ζ134 | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | ζ4ζ1311-ζ4ζ1310-ζ4ζ133+ζ4ζ132 | ζ43ζ139-ζ43ζ137-ζ43ζ136+ζ43ζ134 | ζ4ζ1312-ζ4ζ138-ζ4ζ135+ζ4ζ13 | -ζ43ζ139+ζ43ζ137+ζ43ζ136-ζ43ζ134 | ζ43ζ1312-ζ43ζ138-ζ43ζ135+ζ43ζ13 | -ζ4ζ1311+ζ4ζ1310+ζ4ζ133-ζ4ζ132 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | ζ43ζ1312-ζ43ζ138-ζ43ζ135+ζ43ζ13 | ζ4ζ1311-ζ4ζ1310-ζ4ζ133+ζ4ζ132 | -ζ43ζ139+ζ43ζ137+ζ43ζ136-ζ43ζ134 | -ζ4ζ1311+ζ4ζ1310+ζ4ζ133-ζ4ζ132 | ζ43ζ139-ζ43ζ137-ζ43ζ136+ζ43ζ134 | ζ4ζ1312-ζ4ζ138-ζ4ζ135+ζ4ζ13 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ4ζ1311+ζ4ζ1310+ζ4ζ133-ζ4ζ132 | -ζ43ζ139+ζ43ζ137+ζ43ζ136-ζ43ζ134 | ζ43ζ1312-ζ43ζ138-ζ43ζ135+ζ43ζ13 | ζ43ζ139-ζ43ζ137-ζ43ζ136+ζ43ζ134 | ζ4ζ1312-ζ4ζ138-ζ4ζ135+ζ4ζ13 | ζ4ζ1311-ζ4ζ1310-ζ4ζ133+ζ4ζ132 | complex faithful |
►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 77 40 90 27 103 14 64)(2 72 13 69 28 98 39 95)(3 67 38 100 29 93 12 74)(4 62 11 79 30 88 37 53)(5 57 36 58 31 83 10 84)(6 104 9 89 32 78 35 63)(7 99 34 68 33 73 8 94)(15 59 26 56 41 85 52 82)(16 54 51 87 42 80 25 61)(17 101 24 66 43 75 50 92)(18 96 49 97 44 70 23 71)(19 91 22 76 45 65 48 102)(20 86 47 55 46 60 21 81)
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), (1,77,40,90,27,103,14,64)(2,72,13,69,28,98,39,95)(3,67,38,100,29,93,12,74)(4,62,11,79,30,88,37,53)(5,57,36,58,31,83,10,84)(6,104,9,89,32,78,35,63)(7,99,34,68,33,73,8,94)(15,59,26,56,41,85,52,82)(16,54,51,87,42,80,25,61)(17,101,24,66,43,75,50,92)(18,96,49,97,44,70,23,71)(19,91,22,76,45,65,48,102)(20,86,47,55,46,60,21,81)>;
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), (1,77,40,90,27,103,14,64)(2,72,13,69,28,98,39,95)(3,67,38,100,29,93,12,74)(4,62,11,79,30,88,37,53)(5,57,36,58,31,83,10,84)(6,104,9,89,32,78,35,63)(7,99,34,68,33,73,8,94)(15,59,26,56,41,85,52,82)(16,54,51,87,42,80,25,61)(17,101,24,66,43,75,50,92)(18,96,49,97,44,70,23,71)(19,91,22,76,45,65,48,102)(20,86,47,55,46,60,21,81) );
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)], [(1,77,40,90,27,103,14,64),(2,72,13,69,28,98,39,95),(3,67,38,100,29,93,12,74),(4,62,11,79,30,88,37,53),(5,57,36,58,31,83,10,84),(6,104,9,89,32,78,35,63),(7,99,34,68,33,73,8,94),(15,59,26,56,41,85,52,82),(16,54,51,87,42,80,25,61),(17,101,24,66,43,75,50,92),(18,96,49,97,44,70,23,71),(19,91,22,76,45,65,48,102),(20,86,47,55,46,60,21,81)]])
C52.C4 is a maximal subgroup of
C104.C4 C104.1C4 Dic26⋊C4 D52⋊C4 D13⋊M4(2) Dic26.C4 D52.C4
C52.C4 is a maximal quotient of C52⋊C8 C26.C42 D26⋊C8
Matrix representation of C52.C4 ►in GL6(𝔽313)
| 288 | 0 | 0 | 0 | 0 | 0 |
| 139 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 145 | 176 | 73 |
| 0 | 0 | 240 | 168 | 138 | 241 |
| 0 | 0 | 72 | 103 | 72 | 1 |
| 0 | 0 | 312 | 0 | 0 | 0 |
| 205 | 85 | 0 | 0 | 0 | 0 |
| 3 | 108 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 134 | 113 | 32 |
| 0 | 0 | 42 | 135 | 113 | 103 |
| 0 | 0 | 209 | 168 | 138 | 210 |
| 0 | 0 | 0 | 281 | 272 | 281 |
G:=sub<GL(6,GF(313))| [288,139,0,0,0,0,0,25,0,0,0,0,0,0,103,240,72,312,0,0,145,168,103,0,0,0,176,138,72,0,0,0,73,241,1,0],[205,3,0,0,0,0,85,108,0,0,0,0,0,0,72,42,209,0,0,0,134,135,168,281,0,0,113,113,138,272,0,0,32,103,210,281] >;
C52.C4 in GAP, Magma, Sage, TeX
C_{52}.C_4 % in TeX
G:=Group("C52.C4"); // GroupNames label
G:=SmallGroup(208,29);
// by ID
G=gap.SmallGroup(208,29);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,20,101,46,42,3204,1214]);
// Polycyclic
G:=Group<a,b|a^52=1,b^4=a^26,b*a*b^-1=a^31>;
// generators/relations
Export
Subgroup lattice of C52.C4 in TeX
Character table of C52.C4 in TeX