direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C13×C8⋊C22, D8⋊2C26, C52.63D4, C104⋊7C22, SD16⋊1C26, M4(2)⋊1C26, C52.48C23, C8⋊(C2×C26), C4○D4⋊2C26, D4⋊2(C2×C26), (C13×D8)⋊6C2, (C2×D4)⋊5C26, Q8⋊2(C2×C26), (D4×C26)⋊14C2, (C2×C26).24D4, C26.78(C2×D4), C4.14(D4×C13), C2.15(D4×C26), (C13×SD16)⋊5C2, C4.5(C22×C26), C22.5(D4×C13), (D4×C13)⋊11C22, (C13×M4(2))⋊5C2, (C2×C52).69C22, (Q8×C13)⋊10C22, (C13×C4○D4)⋊7C2, (C2×C4).10(C2×C26), SmallGroup(416,197)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C8⋊C22
G = < a,b,c,d | a13=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >
Subgroups: 116 in 68 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, D4, Q8, C23, C13, M4(2), D8, SD16, C2×D4, C4○D4, C26, C26, C8⋊C22, C52, C52, C2×C26, C2×C26, C104, C2×C52, C2×C52, D4×C13, D4×C13, D4×C13, Q8×C13, C22×C26, C13×M4(2), C13×D8, C13×SD16, D4×C26, C13×C4○D4, C13×C8⋊C22
Quotients: C1, C2, C22, D4, C23, C13, C2×D4, C26, C8⋊C22, C2×C26, D4×C13, C22×C26, D4×C26, C13×C8⋊C22
►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 21 62 95 34 43 86 72)(2 22 63 96 35 44 87 73)(3 23 64 97 36 45 88 74)(4 24 65 98 37 46 89 75)(5 25 53 99 38 47 90 76)(6 26 54 100 39 48 91 77)(7 14 55 101 27 49 79 78)(8 15 56 102 28 50 80 66)(9 16 57 103 29 51 81 67)(10 17 58 104 30 52 82 68)(11 18 59 92 31 40 83 69)(12 19 60 93 32 41 84 70)(13 20 61 94 33 42 85 71)
(14 101)(15 102)(16 103)(17 104)(18 92)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(40 69)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 66)(51 67)(52 68)(53 90)(54 91)(55 79)(56 80)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 88)(65 89)
(14 49)(15 50)(16 51)(17 52)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(66 102)(67 103)(68 104)(69 92)(70 93)(71 94)(72 95)(73 96)(74 97)(75 98)(76 99)(77 100)(78 101)
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), (1,21,62,95,34,43,86,72)(2,22,63,96,35,44,87,73)(3,23,64,97,36,45,88,74)(4,24,65,98,37,46,89,75)(5,25,53,99,38,47,90,76)(6,26,54,100,39,48,91,77)(7,14,55,101,27,49,79,78)(8,15,56,102,28,50,80,66)(9,16,57,103,29,51,81,67)(10,17,58,104,30,52,82,68)(11,18,59,92,31,40,83,69)(12,19,60,93,32,41,84,70)(13,20,61,94,33,42,85,71), (14,101)(15,102)(16,103)(17,104)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,66)(51,67)(52,68)(53,90)(54,91)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89), (14,49)(15,50)(16,51)(17,52)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(66,102)(67,103)(68,104)(69,92)(70,93)(71,94)(72,95)(73,96)(74,97)(75,98)(76,99)(77,100)(78,101)>;
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), (1,21,62,95,34,43,86,72)(2,22,63,96,35,44,87,73)(3,23,64,97,36,45,88,74)(4,24,65,98,37,46,89,75)(5,25,53,99,38,47,90,76)(6,26,54,100,39,48,91,77)(7,14,55,101,27,49,79,78)(8,15,56,102,28,50,80,66)(9,16,57,103,29,51,81,67)(10,17,58,104,30,52,82,68)(11,18,59,92,31,40,83,69)(12,19,60,93,32,41,84,70)(13,20,61,94,33,42,85,71), (14,101)(15,102)(16,103)(17,104)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,66)(51,67)(52,68)(53,90)(54,91)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89), (14,49)(15,50)(16,51)(17,52)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(66,102)(67,103)(68,104)(69,92)(70,93)(71,94)(72,95)(73,96)(74,97)(75,98)(76,99)(77,100)(78,101) );
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)], [(1,21,62,95,34,43,86,72),(2,22,63,96,35,44,87,73),(3,23,64,97,36,45,88,74),(4,24,65,98,37,46,89,75),(5,25,53,99,38,47,90,76),(6,26,54,100,39,48,91,77),(7,14,55,101,27,49,79,78),(8,15,56,102,28,50,80,66),(9,16,57,103,29,51,81,67),(10,17,58,104,30,52,82,68),(11,18,59,92,31,40,83,69),(12,19,60,93,32,41,84,70),(13,20,61,94,33,42,85,71)], [(14,101),(15,102),(16,103),(17,104),(18,92),(19,93),(20,94),(21,95),(22,96),(23,97),(24,98),(25,99),(26,100),(40,69),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,66),(51,67),(52,68),(53,90),(54,91),(55,79),(56,80),(57,81),(58,82),(59,83),(60,84),(61,85),(62,86),(63,87),(64,88),(65,89)], [(14,49),(15,50),(16,51),(17,52),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46),(25,47),(26,48),(66,102),(67,103),(68,104),(69,92),(70,93),(71,94),(72,95),(73,96),(74,97),(75,98),(76,99),(77,100),(78,101)]])
143 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 8A | 8B | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26X | 26Y | ··· | 26BH | 52A | ··· | 52X | 52Y | ··· | 52AJ | 104A | ··· | 104X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
143 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C13 | C26 | C26 | C26 | C26 | C26 | D4 | D4 | D4×C13 | D4×C13 | C8⋊C22 | C13×C8⋊C22 |
| kernel | C13×C8⋊C22 | C13×M4(2) | C13×D8 | C13×SD16 | D4×C26 | C13×C4○D4 | C8⋊C22 | M4(2) | D8 | SD16 | C2×D4 | C4○D4 | C52 | C2×C26 | C4 | C22 | C13 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 12 | 12 | 24 | 24 | 12 | 12 | 1 | 1 | 12 | 12 | 1 | 12 |
Matrix representation of C13×C8⋊C22 ►in GL4(𝔽313) generated by
| 48 | 0 | 0 | 0 |
| 0 | 48 | 0 | 0 |
| 0 | 0 | 48 | 0 |
| 0 | 0 | 0 | 48 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 312 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 |
| 0 | 0 | 0 | 312 |
| 0 | 0 | 312 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 312 | 0 |
| 0 | 0 | 0 | 312 |
G:=sub<GL(4,GF(313))| [48,0,0,0,0,48,0,0,0,0,48,0,0,0,0,48],[0,0,0,1,0,0,1,0,1,0,0,0,0,312,0,0],[1,0,0,0,0,312,0,0,0,0,0,312,0,0,312,0],[1,0,0,0,0,1,0,0,0,0,312,0,0,0,0,312] >;
C13×C8⋊C22 in GAP, Magma, Sage, TeX
C_{13}\times C_8\rtimes C_2^2 % in TeX
G:=Group("C13xC8:C2^2"); // GroupNames label
G:=SmallGroup(416,197);
// by ID
G=gap.SmallGroup(416,197);
# by ID
G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,3818,9364,4690,88]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations