metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52.46D4, C4.11D52, M4(2)⋊3D13, (C2×C4).1D26, (C2×D52).7C2, C52.4C4⋊2C2, C13⋊2(C4.D4), C22.4(C4×D13), C4.21(C13⋊D4), (C13×M4(2))⋊7C2, (C2×C52).13C22, (C22×D13).1C4, C2.9(D26⋊C4), C26.19(C22⋊C4), (C2×C26).22(C2×C4), SmallGroup(416,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.46D4
G = < a,b,c,d | a8=b2=c13=d2=1, bab=a5, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >
2C2
52C2
52C2
26C22
26C22
52C22
52C22
2C26
4D13
4D13
2C8
13C23
13C23
26C8
26D4
26D4
2D26
2D26
4D26
4D26
13M4(2)
13C2×D4
2C104
2D52
2D52
13C4.D4
Smallest permutation representation of C52.46D4
►On 104 points
Generators in S104
(1 88 32 72 21 92 50 63)(2 89 33 73 22 93 51 64)(3 90 34 74 23 94 52 65)(4 91 35 75 24 95 40 53)(5 79 36 76 25 96 41 54)(6 80 37 77 26 97 42 55)(7 81 38 78 14 98 43 56)(8 82 39 66 15 99 44 57)(9 83 27 67 16 100 45 58)(10 84 28 68 17 101 46 59)(11 85 29 69 18 102 47 60)(12 86 30 70 19 103 48 61)(13 87 31 71 20 104 49 62)
(53 75)(54 76)(55 77)(56 78)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(64 73)(65 74)(79 96)(80 97)(81 98)(82 99)(83 100)(84 101)(85 102)(86 103)(87 104)(88 92)(89 93)(90 94)(91 95)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(27 41)(28 40)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(53 59)(54 58)(55 57)(60 65)(61 64)(62 63)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(79 100)(80 99)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 104)(89 103)(90 102)(91 101)
G:=sub<Sym(104)| (1,88,32,72,21,92,50,63)(2,89,33,73,22,93,51,64)(3,90,34,74,23,94,52,65)(4,91,35,75,24,95,40,53)(5,79,36,76,25,96,41,54)(6,80,37,77,26,97,42,55)(7,81,38,78,14,98,43,56)(8,82,39,66,15,99,44,57)(9,83,27,67,16,100,45,58)(10,84,28,68,17,101,46,59)(11,85,29,69,18,102,47,60)(12,86,30,70,19,103,48,61)(13,87,31,71,20,104,49,62), (53,75)(54,76)(55,77)(56,78)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,73)(65,74)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(85,102)(86,103)(87,104)(88,92)(89,93)(90,94)(91,95), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(27,41)(28,40)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(53,59)(54,58)(55,57)(60,65)(61,64)(62,63)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,100)(80,99)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,104)(89,103)(90,102)(91,101)>;
G:=Group( (1,88,32,72,21,92,50,63)(2,89,33,73,22,93,51,64)(3,90,34,74,23,94,52,65)(4,91,35,75,24,95,40,53)(5,79,36,76,25,96,41,54)(6,80,37,77,26,97,42,55)(7,81,38,78,14,98,43,56)(8,82,39,66,15,99,44,57)(9,83,27,67,16,100,45,58)(10,84,28,68,17,101,46,59)(11,85,29,69,18,102,47,60)(12,86,30,70,19,103,48,61)(13,87,31,71,20,104,49,62), (53,75)(54,76)(55,77)(56,78)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,73)(65,74)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(85,102)(86,103)(87,104)(88,92)(89,93)(90,94)(91,95), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(27,41)(28,40)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(53,59)(54,58)(55,57)(60,65)(61,64)(62,63)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,100)(80,99)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,104)(89,103)(90,102)(91,101) );
G=PermutationGroup([[(1,88,32,72,21,92,50,63),(2,89,33,73,22,93,51,64),(3,90,34,74,23,94,52,65),(4,91,35,75,24,95,40,53),(5,79,36,76,25,96,41,54),(6,80,37,77,26,97,42,55),(7,81,38,78,14,98,43,56),(8,82,39,66,15,99,44,57),(9,83,27,67,16,100,45,58),(10,84,28,68,17,101,46,59),(11,85,29,69,18,102,47,60),(12,86,30,70,19,103,48,61),(13,87,31,71,20,104,49,62)], [(53,75),(54,76),(55,77),(56,78),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72),(64,73),(65,74),(79,96),(80,97),(81,98),(82,99),(83,100),(84,101),(85,102),(86,103),(87,104),(88,92),(89,93),(90,94),(91,95)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(27,41),(28,40),(29,52),(30,51),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(53,59),(54,58),(55,57),(60,65),(61,64),(62,63),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(79,100),(80,99),(81,98),(82,97),(83,96),(84,95),(85,94),(86,93),(87,92),(88,104),(89,103),(90,102),(91,101)]])
71 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26L | 52A | ··· | 52L | 52M | ··· | 52R | 104A | ··· | 104X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 52 | 52 | 2 | 2 | 4 | 4 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
71 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D13 | D26 | D52 | C13⋊D4 | C4×D13 | C4.D4 | C52.46D4 |
| kernel | C52.46D4 | C52.4C4 | C13×M4(2) | C2×D52 | C22×D13 | C52 | M4(2) | C2×C4 | C4 | C4 | C22 | C13 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 6 | 6 | 12 | 12 | 12 | 1 | 12 |
Matrix representation of C52.46D4 ►in GL4(𝔽313) generated by
| 155 | 296 | 311 | 0 |
| 90 | 221 | 0 | 311 |
| 59 | 137 | 158 | 17 |
| 222 | 101 | 223 | 92 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 155 | 296 | 312 | 0 |
| 90 | 221 | 0 | 312 |
| 6 | 1 | 0 | 0 |
| 277 | 255 | 0 | 0 |
| 261 | 198 | 6 | 1 |
| 127 | 52 | 277 | 255 |
| 235 | 37 | 0 | 0 |
| 64 | 78 | 0 | 0 |
| 295 | 270 | 235 | 37 |
| 100 | 18 | 64 | 78 |
G:=sub<GL(4,GF(313))| [155,90,59,222,296,221,137,101,311,0,158,223,0,311,17,92],[1,0,155,90,0,1,296,221,0,0,312,0,0,0,0,312],[6,277,261,127,1,255,198,52,0,0,6,277,0,0,1,255],[235,64,295,100,37,78,270,18,0,0,235,64,0,0,37,78] >;
C52.46D4 in GAP, Magma, Sage, TeX
C_{52}._{46}D_4 % in TeX
G:=Group("C52.46D4"); // GroupNames label
G:=SmallGroup(416,30);
// by ID
G=gap.SmallGroup(416,30);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,121,31,362,86,297,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^13=d^2=1,b*a*b=a^5,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export