direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4×D13, C52⋊C23, C23⋊3D26, D52⋊7C22, D26⋊2C23, C26.5C24, Dic13⋊1C23, C26⋊2(C2×D4), (C2×C4)⋊6D26, (C2×C26)⋊C23, (D4×C26)⋊5C2, C13⋊2(C22×D4), (C2×D52)⋊11C2, (C2×C52)⋊2C22, C4⋊1(C22×D13), (C4×D13)⋊3C22, (D4×C13)⋊5C22, (C23×D13)⋊4C2, C13⋊D4⋊1C22, C2.6(C23×D13), (C22×C26)⋊4C22, C22⋊1(C22×D13), (C2×Dic13)⋊8C22, (C22×D13)⋊6C22, (C2×C4×D13)⋊3C2, (C2×C13⋊D4)⋊9C2, SmallGroup(416,216)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4×D13
G = < a,b,c,d,e | a2=b4=c2=d13=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1792 in 236 conjugacy classes, 97 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, D4, C23, C23, C13, C22×C4, C2×D4, C2×D4, C24, D13, D13, C26, C26, C26, C22×D4, Dic13, C52, D26, D26, C2×C26, C2×C26, C2×C26, C4×D13, D52, C2×Dic13, C13⋊D4, C2×C52, D4×C13, C22×D13, C22×D13, C22×D13, C22×C26, C2×C4×D13, C2×D52, D4×D13, C2×C13⋊D4, D4×C26, C23×D13, C2×D4×D13
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, D13, C22×D4, D26, C22×D13, D4×D13, C23×D13, C2×D4×D13
►On 104 points
Generators in S104
(1 65)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 66)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 79)(38 80)(39 81)(40 98)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)
(1 102 18 90)(2 103 19 91)(3 104 20 79)(4 92 21 80)(5 93 22 81)(6 94 23 82)(7 95 24 83)(8 96 25 84)(9 97 26 85)(10 98 14 86)(11 99 15 87)(12 100 16 88)(13 101 17 89)(27 57 49 69)(28 58 50 70)(29 59 51 71)(30 60 52 72)(31 61 40 73)(32 62 41 74)(33 63 42 75)(34 64 43 76)(35 65 44 77)(36 53 45 78)(37 54 46 66)(38 55 47 67)(39 56 48 68)
(1 90)(2 91)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 87)(12 88)(13 89)(14 98)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 53)(37 54)(38 55)(39 56)(40 73)(41 74)(42 75)(43 76)(44 77)(45 78)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)
(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 76)(2 75)(3 74)(4 73)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 78)(13 77)(14 55)(15 54)(16 53)(17 65)(18 64)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 57)(26 56)(27 96)(28 95)(29 94)(30 93)(31 92)(32 104)(33 103)(34 102)(35 101)(36 100)(37 99)(38 98)(39 97)(40 80)(41 79)(42 91)(43 90)(44 89)(45 88)(46 87)(47 86)(48 85)(49 84)(50 83)(51 82)(52 81)
G:=sub<Sym(104)| (1,65)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,66)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,79)(38,80)(39,81)(40,98)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97), (1,102,18,90)(2,103,19,91)(3,104,20,79)(4,92,21,80)(5,93,22,81)(6,94,23,82)(7,95,24,83)(8,96,25,84)(9,97,26,85)(10,98,14,86)(11,99,15,87)(12,100,16,88)(13,101,17,89)(27,57,49,69)(28,58,50,70)(29,59,51,71)(30,60,52,72)(31,61,40,73)(32,62,41,74)(33,63,42,75)(34,64,43,76)(35,65,44,77)(36,53,45,78)(37,54,46,66)(38,55,47,67)(39,56,48,68), (1,90)(2,91)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,53)(37,54)(38,55)(39,56)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72), (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), (1,76)(2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,78)(13,77)(14,55)(15,54)(16,53)(17,65)(18,64)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,57)(26,56)(27,96)(28,95)(29,94)(30,93)(31,92)(32,104)(33,103)(34,102)(35,101)(36,100)(37,99)(38,98)(39,97)(40,80)(41,79)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)>;
G:=Group( (1,65)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,66)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,79)(38,80)(39,81)(40,98)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97), (1,102,18,90)(2,103,19,91)(3,104,20,79)(4,92,21,80)(5,93,22,81)(6,94,23,82)(7,95,24,83)(8,96,25,84)(9,97,26,85)(10,98,14,86)(11,99,15,87)(12,100,16,88)(13,101,17,89)(27,57,49,69)(28,58,50,70)(29,59,51,71)(30,60,52,72)(31,61,40,73)(32,62,41,74)(33,63,42,75)(34,64,43,76)(35,65,44,77)(36,53,45,78)(37,54,46,66)(38,55,47,67)(39,56,48,68), (1,90)(2,91)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,53)(37,54)(38,55)(39,56)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72), (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), (1,76)(2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,78)(13,77)(14,55)(15,54)(16,53)(17,65)(18,64)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,57)(26,56)(27,96)(28,95)(29,94)(30,93)(31,92)(32,104)(33,103)(34,102)(35,101)(36,100)(37,99)(38,98)(39,97)(40,80)(41,79)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81) );
G=PermutationGroup([[(1,65),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,66),(21,67),(22,68),(23,69),(24,70),(25,71),(26,72),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,79),(38,80),(39,81),(40,98),(41,99),(42,100),(43,101),(44,102),(45,103),(46,104),(47,92),(48,93),(49,94),(50,95),(51,96),(52,97)], [(1,102,18,90),(2,103,19,91),(3,104,20,79),(4,92,21,80),(5,93,22,81),(6,94,23,82),(7,95,24,83),(8,96,25,84),(9,97,26,85),(10,98,14,86),(11,99,15,87),(12,100,16,88),(13,101,17,89),(27,57,49,69),(28,58,50,70),(29,59,51,71),(30,60,52,72),(31,61,40,73),(32,62,41,74),(33,63,42,75),(34,64,43,76),(35,65,44,77),(36,53,45,78),(37,54,46,66),(38,55,47,67),(39,56,48,68)], [(1,90),(2,91),(3,79),(4,80),(5,81),(6,82),(7,83),(8,84),(9,85),(10,86),(11,87),(12,88),(13,89),(14,98),(15,99),(16,100),(17,101),(18,102),(19,103),(20,104),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,63),(34,64),(35,65),(36,53),(37,54),(38,55),(39,56),(40,73),(41,74),(42,75),(43,76),(44,77),(45,78),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72)], [(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)], [(1,76),(2,75),(3,74),(4,73),(5,72),(6,71),(7,70),(8,69),(9,68),(10,67),(11,66),(12,78),(13,77),(14,55),(15,54),(16,53),(17,65),(18,64),(19,63),(20,62),(21,61),(22,60),(23,59),(24,58),(25,57),(26,56),(27,96),(28,95),(29,94),(30,93),(31,92),(32,104),(33,103),(34,102),(35,101),(36,100),(37,99),(38,98),(39,97),(40,80),(41,79),(42,91),(43,90),(44,89),(45,88),(46,87),(47,86),(48,85),(49,84),(50,83),(51,82),(52,81)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 13A | ··· | 13F | 26A | ··· | 26R | 26S | ··· | 26AP | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 13 | 13 | 13 | 13 | 26 | 26 | 26 | 26 | 2 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D13 | D26 | D26 | D26 | D4×D13 |
| kernel | C2×D4×D13 | C2×C4×D13 | C2×D52 | D4×D13 | C2×C13⋊D4 | D4×C26 | C23×D13 | D26 | C2×D4 | C2×C4 | D4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 4 | 6 | 6 | 24 | 12 | 12 |
Matrix representation of C2×D4×D13 ►in GL4(𝔽53) generated by
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 0 | 52 |
| 0 | 0 | 1 | 0 |
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 0 | 52 |
| 0 | 0 | 52 | 0 |
| 41 | 1 | 0 | 0 |
| 24 | 20 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 45 | 0 | 0 |
| 33 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,52,0,0,0,0,0,1,0,0,52,0],[52,0,0,0,0,52,0,0,0,0,0,52,0,0,52,0],[41,24,0,0,1,20,0,0,0,0,1,0,0,0,0,1],[0,33,0,0,45,0,0,0,0,0,1,0,0,0,0,1] >;
C2×D4×D13 in GAP, Magma, Sage, TeX
C_2\times D_4\times D_{13} % in TeX
G:=Group("C2xD4xD13"); // GroupNames label
G:=SmallGroup(416,216);
// by ID
G=gap.SmallGroup(416,216);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,159,13829]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^13=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations