direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×M4(2), C4.C52, C8⋊3C26, C104⋊7C2, C52.7C4, C22.C52, C52.22C22, (C2×C4).2C26, C2.3(C2×C52), (C2×C52).8C2, C4.6(C2×C26), (C2×C26).3C4, C26.19(C2×C4), SmallGroup(208,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×M4(2)
G = < a,b,c | a13=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Smallest permutation representation of C13×M4(2)
►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 26 80 39 57 48 67 94)(2 14 81 27 58 49 68 95)(3 15 82 28 59 50 69 96)(4 16 83 29 60 51 70 97)(5 17 84 30 61 52 71 98)(6 18 85 31 62 40 72 99)(7 19 86 32 63 41 73 100)(8 20 87 33 64 42 74 101)(9 21 88 34 65 43 75 102)(10 22 89 35 53 44 76 103)(11 23 90 36 54 45 77 104)(12 24 91 37 55 46 78 92)(13 25 79 38 56 47 66 93)
(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)(27 95)(28 96)(29 97)(30 98)(31 99)(32 100)(33 101)(34 102)(35 103)(36 104)(37 92)(38 93)(39 94)
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), (1,26,80,39,57,48,67,94)(2,14,81,27,58,49,68,95)(3,15,82,28,59,50,69,96)(4,16,83,29,60,51,70,97)(5,17,84,30,61,52,71,98)(6,18,85,31,62,40,72,99)(7,19,86,32,63,41,73,100)(8,20,87,33,64,42,74,101)(9,21,88,34,65,43,75,102)(10,22,89,35,53,44,76,103)(11,23,90,36,54,45,77,104)(12,24,91,37,55,46,78,92)(13,25,79,38,56,47,66,93), (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)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,92)(38,93)(39,94)>;
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), (1,26,80,39,57,48,67,94)(2,14,81,27,58,49,68,95)(3,15,82,28,59,50,69,96)(4,16,83,29,60,51,70,97)(5,17,84,30,61,52,71,98)(6,18,85,31,62,40,72,99)(7,19,86,32,63,41,73,100)(8,20,87,33,64,42,74,101)(9,21,88,34,65,43,75,102)(10,22,89,35,53,44,76,103)(11,23,90,36,54,45,77,104)(12,24,91,37,55,46,78,92)(13,25,79,38,56,47,66,93), (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)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,92)(38,93)(39,94) );
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)], [(1,26,80,39,57,48,67,94),(2,14,81,27,58,49,68,95),(3,15,82,28,59,50,69,96),(4,16,83,29,60,51,70,97),(5,17,84,30,61,52,71,98),(6,18,85,31,62,40,72,99),(7,19,86,32,63,41,73,100),(8,20,87,33,64,42,74,101),(9,21,88,34,65,43,75,102),(10,22,89,35,53,44,76,103),(11,23,90,36,54,45,77,104),(12,24,91,37,55,46,78,92),(13,25,79,38,56,47,66,93)], [(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),(27,95),(28,96),(29,97),(30,98),(31,99),(32,100),(33,101),(34,102),(35,103),(36,104),(37,92),(38,93),(39,94)]])
C13×M4(2) is a maximal subgroup of
C52.53D4 C52.46D4 C4.12D52 D52⋊7C4 D52.2C4 C8⋊D26 C8.D26
130 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26X | 52A | ··· | 52X | 52Y | ··· | 52AJ | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C13 | C26 | C26 | C52 | C52 | M4(2) | C13×M4(2) |
| kernel | C13×M4(2) | C104 | C2×C52 | C52 | C2×C26 | M4(2) | C8 | C2×C4 | C4 | C22 | C13 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 12 | 24 | 12 | 24 | 24 | 2 | 24 |
Matrix representation of C13×M4(2) ►in GL2(𝔽313) generated by
| 280 | 0 |
| 0 | 280 |
| 253 | 311 |
| 66 | 60 |
| 1 | 0 |
| 253 | 312 |
G:=sub<GL(2,GF(313))| [280,0,0,280],[253,66,311,60],[1,253,0,312] >;
C13×M4(2) in GAP, Magma, Sage, TeX
C_{13}\times M_4(2) % in TeX
G:=Group("C13xM4(2)"); // GroupNames label
G:=SmallGroup(208,24);
// by ID
G=gap.SmallGroup(208,24);
# by ID
G:=PCGroup([5,-2,-2,-13,-2,-2,260,1061,58]);
// Polycyclic
G:=Group<a,b,c|a^13=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations
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