direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C13×C23⋊C4, C23⋊C52, (C2×C4)⋊C52, (C2×C52)⋊7C4, C22⋊C4⋊1C26, (C22×C26)⋊1C4, (C2×D4).1C26, (D4×C26).7C2, (C2×C26).21D4, C22.2(C2×C52), C23.1(C2×C26), C22.2(D4×C13), C26.32(C22⋊C4), (C22×C26).1C22, (C13×C22⋊C4)⋊2C2, (C2×C26).39(C2×C4), C2.3(C13×C22⋊C4), SmallGroup(416,49)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C23⋊C4
G = < a,b,c,d,e | a13=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >
Smallest permutation representation of C13×C23⋊C4
►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)(2 78)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(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 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 79)(39 80)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 53)
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 79)(9 80)(10 81)(11 82)(12 83)(13 84)(14 48)(15 49)(16 50)(17 51)(18 52)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(53 102)(54 103)(55 104)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)
(1 50)(2 51)(3 52)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 90)(22 91)(23 79)(24 80)(25 81)(26 82)(27 96)(28 97)(29 98)(30 99)(31 100)(32 101)(33 102)(34 103)(35 104)(36 92)(37 93)(38 94)(39 95)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)
(1 77 85 100)(2 78 86 101)(3 66 87 102)(4 67 88 103)(5 68 89 104)(6 69 90 92)(7 70 91 93)(8 71 79 94)(9 72 80 95)(10 73 81 96)(11 74 82 97)(12 75 83 98)(13 76 84 99)(14 29 48 62)(15 30 49 63)(16 31 50 64)(17 32 51 65)(18 33 52 53)(19 34 40 54)(20 35 41 55)(21 36 42 56)(22 37 43 57)(23 38 44 58)(24 39 45 59)(25 27 46 60)(26 28 47 61)
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)(2,78)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(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)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,79)(39,80)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,53), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,48)(15,49)(16,50)(17,51)(18,52)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(53,102)(54,103)(55,104)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101), (1,50)(2,51)(3,52)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,91)(23,79)(24,80)(25,81)(26,82)(27,96)(28,97)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,92)(37,93)(38,94)(39,95)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78), (1,77,85,100)(2,78,86,101)(3,66,87,102)(4,67,88,103)(5,68,89,104)(6,69,90,92)(7,70,91,93)(8,71,79,94)(9,72,80,95)(10,73,81,96)(11,74,82,97)(12,75,83,98)(13,76,84,99)(14,29,48,62)(15,30,49,63)(16,31,50,64)(17,32,51,65)(18,33,52,53)(19,34,40,54)(20,35,41,55)(21,36,42,56)(22,37,43,57)(23,38,44,58)(24,39,45,59)(25,27,46,60)(26,28,47,61)>;
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)(2,78)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(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)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,79)(39,80)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,53), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,48)(15,49)(16,50)(17,51)(18,52)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(53,102)(54,103)(55,104)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101), (1,50)(2,51)(3,52)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,91)(23,79)(24,80)(25,81)(26,82)(27,96)(28,97)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,92)(37,93)(38,94)(39,95)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78), (1,77,85,100)(2,78,86,101)(3,66,87,102)(4,67,88,103)(5,68,89,104)(6,69,90,92)(7,70,91,93)(8,71,79,94)(9,72,80,95)(10,73,81,96)(11,74,82,97)(12,75,83,98)(13,76,84,99)(14,29,48,62)(15,30,49,63)(16,31,50,64)(17,32,51,65)(18,33,52,53)(19,34,40,54)(20,35,41,55)(21,36,42,56)(22,37,43,57)(23,38,44,58)(24,39,45,59)(25,27,46,60)(26,28,47,61) );
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),(2,78),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(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),(27,81),(28,82),(29,83),(30,84),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,91),(38,79),(39,80),(40,54),(41,55),(42,56),(43,57),(44,58),(45,59),(46,60),(47,61),(48,62),(49,63),(50,64),(51,65),(52,53)], [(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,79),(9,80),(10,81),(11,82),(12,83),(13,84),(14,48),(15,49),(16,50),(17,51),(18,52),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,66),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(53,102),(54,103),(55,104),(56,92),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,99),(64,100),(65,101)], [(1,50),(2,51),(3,52),(4,40),(5,41),(6,42),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,49),(14,83),(15,84),(16,85),(17,86),(18,87),(19,88),(20,89),(21,90),(22,91),(23,79),(24,80),(25,81),(26,82),(27,96),(28,97),(29,98),(30,99),(31,100),(32,101),(33,102),(34,103),(35,104),(36,92),(37,93),(38,94),(39,95),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,73),(61,74),(62,75),(63,76),(64,77),(65,78)], [(1,77,85,100),(2,78,86,101),(3,66,87,102),(4,67,88,103),(5,68,89,104),(6,69,90,92),(7,70,91,93),(8,71,79,94),(9,72,80,95),(10,73,81,96),(11,74,82,97),(12,75,83,98),(13,76,84,99),(14,29,48,62),(15,30,49,63),(16,31,50,64),(17,32,51,65),(18,33,52,53),(19,34,40,54),(20,35,41,55),(21,36,42,56),(22,37,43,57),(23,38,44,58),(24,39,45,59),(25,27,46,60),(26,28,47,61)]])
143 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4E | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26AV | 26AW | ··· | 26BH | 52A | ··· | 52BH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
143 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C13 | C26 | C26 | C52 | C52 | D4 | D4×C13 | C23⋊C4 | C13×C23⋊C4 |
| kernel | C13×C23⋊C4 | C13×C22⋊C4 | D4×C26 | C2×C52 | C22×C26 | C23⋊C4 | C22⋊C4 | C2×D4 | C2×C4 | C23 | C2×C26 | C22 | C13 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 12 | 24 | 12 | 24 | 24 | 2 | 24 | 1 | 12 |
Matrix representation of C13×C23⋊C4 ►in GL4(𝔽53) generated by
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 20 | 0 | 51 | 0 |
| 7 | 0 | 52 | 1 |
| 14 | 0 | 33 | 0 |
| 33 | 1 | 47 | 0 |
| 1 | 51 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 7 | 33 | 0 | 1 |
| 46 | 47 | 1 | 0 |
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 47 | 0 | 0 | 2 |
| 40 | 0 | 1 | 1 |
| 19 | 0 | 0 | 20 |
| 9 | 52 | 0 | 6 |
G:=sub<GL(4,GF(53))| [46,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[20,7,14,33,0,0,0,1,51,52,33,47,0,1,0,0],[1,0,7,46,51,52,33,47,0,0,0,1,0,0,1,0],[52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[47,40,19,9,0,0,0,52,0,1,0,0,2,1,20,6] >;
C13×C23⋊C4 in GAP, Magma, Sage, TeX
C_{13}\times C_2^3\rtimes C_4 % in TeX
G:=Group("C13xC2^3:C4"); // GroupNames label
G:=SmallGroup(416,49);
// by ID
G=gap.SmallGroup(416,49);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,6243,4690]);
// Polycyclic
G:=Group<a,b,c,d,e|a^13=b^2=c^2=d^2=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations
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