direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×C4○D4, D4⋊2C26, Q8⋊2C26, C52.21C22, C26.13C23, (C2×C52)⋊7C2, (C2×C4)⋊3C26, C52○(D4×C13), C52○(Q8×C13), (D4×C13)⋊5C2, C4.5(C2×C26), (Q8×C13)⋊5C2, C22.(C2×C26), (C2×C26).2C22, C2.3(C22×C26), SmallGroup(208,48)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C4○D4
G = < a,b,c,d | a13=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Smallest permutation representation of C13×C4○D4
►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 58 50 101)(2 59 51 102)(3 60 52 103)(4 61 40 104)(5 62 41 92)(6 63 42 93)(7 64 43 94)(8 65 44 95)(9 53 45 96)(10 54 46 97)(11 55 47 98)(12 56 48 99)(13 57 49 100)(14 81 69 35)(15 82 70 36)(16 83 71 37)(17 84 72 38)(18 85 73 39)(19 86 74 27)(20 87 75 28)(21 88 76 29)(22 89 77 30)(23 90 78 31)(24 91 66 32)(25 79 67 33)(26 80 68 34)
(1 101 50 58)(2 102 51 59)(3 103 52 60)(4 104 40 61)(5 92 41 62)(6 93 42 63)(7 94 43 64)(8 95 44 65)(9 96 45 53)(10 97 46 54)(11 98 47 55)(12 99 48 56)(13 100 49 57)(14 81 69 35)(15 82 70 36)(16 83 71 37)(17 84 72 38)(18 85 73 39)(19 86 74 27)(20 87 75 28)(21 88 76 29)(22 89 77 30)(23 90 78 31)(24 91 66 32)(25 79 67 33)(26 80 68 34)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 59)(15 60)(16 61)(17 62)(18 63)(19 64)(20 65)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 79)(50 80)(51 81)(52 82)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 92)(73 93)(74 94)(75 95)(76 96)(77 97)(78 98)
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), (1,58,50,101)(2,59,51,102)(3,60,52,103)(4,61,40,104)(5,62,41,92)(6,63,42,93)(7,64,43,94)(8,65,44,95)(9,53,45,96)(10,54,46,97)(11,55,47,98)(12,56,48,99)(13,57,49,100)(14,81,69,35)(15,82,70,36)(16,83,71,37)(17,84,72,38)(18,85,73,39)(19,86,74,27)(20,87,75,28)(21,88,76,29)(22,89,77,30)(23,90,78,31)(24,91,66,32)(25,79,67,33)(26,80,68,34), (1,101,50,58)(2,102,51,59)(3,103,52,60)(4,104,40,61)(5,92,41,62)(6,93,42,63)(7,94,43,64)(8,95,44,65)(9,96,45,53)(10,97,46,54)(11,98,47,55)(12,99,48,56)(13,100,49,57)(14,81,69,35)(15,82,70,36)(16,83,71,37)(17,84,72,38)(18,85,73,39)(19,86,74,27)(20,87,75,28)(21,88,76,29)(22,89,77,30)(23,90,78,31)(24,91,66,32)(25,79,67,33)(26,80,68,34), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,79)(50,80)(51,81)(52,82)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)>;
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), (1,58,50,101)(2,59,51,102)(3,60,52,103)(4,61,40,104)(5,62,41,92)(6,63,42,93)(7,64,43,94)(8,65,44,95)(9,53,45,96)(10,54,46,97)(11,55,47,98)(12,56,48,99)(13,57,49,100)(14,81,69,35)(15,82,70,36)(16,83,71,37)(17,84,72,38)(18,85,73,39)(19,86,74,27)(20,87,75,28)(21,88,76,29)(22,89,77,30)(23,90,78,31)(24,91,66,32)(25,79,67,33)(26,80,68,34), (1,101,50,58)(2,102,51,59)(3,103,52,60)(4,104,40,61)(5,92,41,62)(6,93,42,63)(7,94,43,64)(8,95,44,65)(9,96,45,53)(10,97,46,54)(11,98,47,55)(12,99,48,56)(13,100,49,57)(14,81,69,35)(15,82,70,36)(16,83,71,37)(17,84,72,38)(18,85,73,39)(19,86,74,27)(20,87,75,28)(21,88,76,29)(22,89,77,30)(23,90,78,31)(24,91,66,32)(25,79,67,33)(26,80,68,34), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,79)(50,80)(51,81)(52,82)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98) );
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)], [(1,58,50,101),(2,59,51,102),(3,60,52,103),(4,61,40,104),(5,62,41,92),(6,63,42,93),(7,64,43,94),(8,65,44,95),(9,53,45,96),(10,54,46,97),(11,55,47,98),(12,56,48,99),(13,57,49,100),(14,81,69,35),(15,82,70,36),(16,83,71,37),(17,84,72,38),(18,85,73,39),(19,86,74,27),(20,87,75,28),(21,88,76,29),(22,89,77,30),(23,90,78,31),(24,91,66,32),(25,79,67,33),(26,80,68,34)], [(1,101,50,58),(2,102,51,59),(3,103,52,60),(4,104,40,61),(5,92,41,62),(6,93,42,63),(7,94,43,64),(8,95,44,65),(9,96,45,53),(10,97,46,54),(11,98,47,55),(12,99,48,56),(13,100,49,57),(14,81,69,35),(15,82,70,36),(16,83,71,37),(17,84,72,38),(18,85,73,39),(19,86,74,27),(20,87,75,28),(21,88,76,29),(22,89,77,30),(23,90,78,31),(24,91,66,32),(25,79,67,33),(26,80,68,34)], [(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,59),(15,60),(16,61),(17,62),(18,63),(19,64),(20,65),(21,53),(22,54),(23,55),(24,56),(25,57),(26,58),(40,83),(41,84),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,79),(50,80),(51,81),(52,82),(66,99),(67,100),(68,101),(69,102),(70,103),(71,104),(72,92),(73,93),(74,94),(75,95),(76,96),(77,97),(78,98)]])
C13×C4○D4 is a maximal subgroup of
C52.56D4 D4.Dic13 D4⋊D26 C52.C23 D4.9D26 D4⋊8D26 D4.10D26
C13×C4○D4 is a maximal quotient of D4×C52 Q8×C52
130 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26AV | 52A | ··· | 52X | 52Y | ··· | 52BH |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C13 | C26 | C26 | C26 | C4○D4 | C13×C4○D4 |
| kernel | C13×C4○D4 | C2×C52 | D4×C13 | Q8×C13 | C4○D4 | C2×C4 | D4 | Q8 | C13 | C1 |
| # reps | 1 | 3 | 3 | 1 | 12 | 36 | 36 | 12 | 2 | 24 |
Matrix representation of C13×C4○D4 ►in GL2(𝔽53) generated by
| 46 | 0 |
| 0 | 46 |
| 23 | 0 |
| 0 | 23 |
| 30 | 0 |
| 21 | 23 |
| 37 | 23 |
| 35 | 16 |
G:=sub<GL(2,GF(53))| [46,0,0,46],[23,0,0,23],[30,21,0,23],[37,35,23,16] >;
C13×C4○D4 in GAP, Magma, Sage, TeX
C_{13}\times C_4\circ D_4 % in TeX
G:=Group("C13xC4oD4"); // GroupNames label
G:=SmallGroup(208,48);
// by ID
G=gap.SmallGroup(208,48);
# by ID
G:=PCGroup([5,-2,-2,-2,-13,-2,1061,402]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations
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