direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C13×C4≀C2, D4⋊2C52, Q8⋊2C52, C42⋊3C26, C52.66D4, M4(2)⋊4C26, (C4×C52)⋊10C2, (D4×C13)⋊8C4, C4.3(C2×C52), (Q8×C13)⋊8C4, C52.51(C2×C4), C4○D4.1C26, C4.17(D4×C13), (C2×C26).22D4, C22.3(D4×C13), C26.37(C22⋊C4), (C13×M4(2))⋊10C2, (C2×C52).116C22, (C2×C4).19(C2×C26), (C13×C4○D4).4C2, C2.8(C13×C22⋊C4), SmallGroup(416,54)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C4≀C2
G = < a,b,c,d | a13=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >
Smallest permutation representation of C13×C4≀C2
►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 39 22 86)(2 27 23 87)(3 28 24 88)(4 29 25 89)(5 30 26 90)(6 31 14 91)(7 32 15 79)(8 33 16 80)(9 34 17 81)(10 35 18 82)(11 36 19 83)(12 37 20 84)(13 38 21 85)(40 55 70 95)(41 56 71 96)(42 57 72 97)(43 58 73 98)(44 59 74 99)(45 60 75 100)(46 61 76 101)(47 62 77 102)(48 63 78 103)(49 64 66 104)(50 65 67 92)(51 53 68 93)(52 54 69 94)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 92)(11 93)(12 94)(13 95)(14 61)(15 62)(16 63)(17 64)(18 65)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)
(1 86 22 39)(2 87 23 27)(3 88 24 28)(4 89 25 29)(5 90 26 30)(6 91 14 31)(7 79 15 32)(8 80 16 33)(9 81 17 34)(10 82 18 35)(11 83 19 36)(12 84 20 37)(13 85 21 38)
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), (1,39,22,86)(2,27,23,87)(3,28,24,88)(4,29,25,89)(5,30,26,90)(6,31,14,91)(7,32,15,79)(8,33,16,80)(9,34,17,81)(10,35,18,82)(11,36,19,83)(12,37,20,84)(13,38,21,85)(40,55,70,95)(41,56,71,96)(42,57,72,97)(43,58,73,98)(44,59,74,99)(45,60,75,100)(46,61,76,101)(47,62,77,102)(48,63,78,103)(49,64,66,104)(50,65,67,92)(51,53,68,93)(52,54,69,94), (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,92)(11,93)(12,94)(13,95)(14,61)(15,62)(16,63)(17,64)(18,65)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84), (1,86,22,39)(2,87,23,27)(3,88,24,28)(4,89,25,29)(5,90,26,30)(6,91,14,31)(7,79,15,32)(8,80,16,33)(9,81,17,34)(10,82,18,35)(11,83,19,36)(12,84,20,37)(13,85,21,38)>;
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), (1,39,22,86)(2,27,23,87)(3,28,24,88)(4,29,25,89)(5,30,26,90)(6,31,14,91)(7,32,15,79)(8,33,16,80)(9,34,17,81)(10,35,18,82)(11,36,19,83)(12,37,20,84)(13,38,21,85)(40,55,70,95)(41,56,71,96)(42,57,72,97)(43,58,73,98)(44,59,74,99)(45,60,75,100)(46,61,76,101)(47,62,77,102)(48,63,78,103)(49,64,66,104)(50,65,67,92)(51,53,68,93)(52,54,69,94), (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,92)(11,93)(12,94)(13,95)(14,61)(15,62)(16,63)(17,64)(18,65)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84), (1,86,22,39)(2,87,23,27)(3,88,24,28)(4,89,25,29)(5,90,26,30)(6,91,14,31)(7,79,15,32)(8,80,16,33)(9,81,17,34)(10,82,18,35)(11,83,19,36)(12,84,20,37)(13,85,21,38) );
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)], [(1,39,22,86),(2,27,23,87),(3,28,24,88),(4,29,25,89),(5,30,26,90),(6,31,14,91),(7,32,15,79),(8,33,16,80),(9,34,17,81),(10,35,18,82),(11,36,19,83),(12,37,20,84),(13,38,21,85),(40,55,70,95),(41,56,71,96),(42,57,72,97),(43,58,73,98),(44,59,74,99),(45,60,75,100),(46,61,76,101),(47,62,77,102),(48,63,78,103),(49,64,66,104),(50,65,67,92),(51,53,68,93),(52,54,69,94)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,101),(7,102),(8,103),(9,104),(10,92),(11,93),(12,94),(13,95),(14,61),(15,62),(16,63),(17,64),(18,65),(19,53),(20,54),(21,55),(22,56),(23,57),(24,58),(25,59),(26,60),(27,72),(28,73),(29,74),(30,75),(31,76),(32,77),(33,78),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,85),(41,86),(42,87),(43,88),(44,89),(45,90),(46,91),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84)], [(1,86,22,39),(2,87,23,27),(3,88,24,28),(4,89,25,29),(5,90,26,30),(6,91,14,31),(7,79,15,32),(8,80,16,33),(9,81,17,34),(10,82,18,35),(11,83,19,36),(12,84,20,37),(13,85,21,38)]])
182 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | ··· | 4G | 4H | 8A | 8B | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26X | 26Y | ··· | 26AJ | 52A | ··· | 52X | 52Y | ··· | 52CF | 52CG | ··· | 52CR | 104A | ··· | 104X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
182 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C13 | C26 | C26 | C26 | C52 | C52 | D4 | D4 | C4≀C2 | D4×C13 | D4×C13 | C13×C4≀C2 |
| kernel | C13×C4≀C2 | C4×C52 | C13×M4(2) | C13×C4○D4 | D4×C13 | Q8×C13 | C4≀C2 | C42 | M4(2) | C4○D4 | D4 | Q8 | C52 | C2×C26 | C13 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 12 | 12 | 24 | 24 | 1 | 1 | 4 | 12 | 12 | 48 |
Matrix representation of C13×C4≀C2 ►in GL2(𝔽313) generated by
| 277 | 0 |
| 0 | 277 |
| 288 | 0 |
| 0 | 25 |
| 0 | 25 |
| 288 | 0 |
| 25 | 0 |
| 0 | 1 |
G:=sub<GL(2,GF(313))| [277,0,0,277],[288,0,0,25],[0,288,25,0],[25,0,0,1] >;
C13×C4≀C2 in GAP, Magma, Sage, TeX
C_{13}\times C_4\wr C_2 % in TeX
G:=Group("C13xC4wrC2"); // GroupNames label
G:=SmallGroup(416,54);
// by ID
G=gap.SmallGroup(416,54);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,6243,3129,117,88]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;
// generators/relations
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