direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×C22⋊C4, C22⋊C52, C23.C26, C26.12D4, (C2×C26)⋊3C4, (C2×C52)⋊2C2, (C2×C4)⋊1C26, C2.1(C2×C52), C2.1(D4×C13), C26.17(C2×C4), C22.2(C2×C26), (C22×C26).1C2, (C2×C26).13C22, SmallGroup(208,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C22⋊C4
G = < a,b,c,d | a13=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Smallest permutation representation of C13×C22⋊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 65)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 91)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(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)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 102)(15 103)(16 104)(17 92)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)(46 63)(47 64)(48 65)(49 53)(50 54)(51 55)(52 56)(66 91)(67 79)(68 80)(69 81)(70 82)(71 83)(72 84)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)
(1 67 65 15)(2 68 53 16)(3 69 54 17)(4 70 55 18)(5 71 56 19)(6 72 57 20)(7 73 58 21)(8 74 59 22)(9 75 60 23)(10 76 61 24)(11 77 62 25)(12 78 63 26)(13 66 64 14)(27 84 40 95)(28 85 41 96)(29 86 42 97)(30 87 43 98)(31 88 44 99)(32 89 45 100)(33 90 46 101)(34 91 47 102)(35 79 48 103)(36 80 49 104)(37 81 50 92)(38 82 51 93)(39 83 52 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), (1,65)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,91)(15,79)(16,80)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(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), (1,35)(2,36)(3,37)(4,38)(5,39)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,102)(15,103)(16,104)(17,92)(18,93)(19,94)(20,95)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,65)(49,53)(50,54)(51,55)(52,56)(66,91)(67,79)(68,80)(69,81)(70,82)(71,83)(72,84)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90), (1,67,65,15)(2,68,53,16)(3,69,54,17)(4,70,55,18)(5,71,56,19)(6,72,57,20)(7,73,58,21)(8,74,59,22)(9,75,60,23)(10,76,61,24)(11,77,62,25)(12,78,63,26)(13,66,64,14)(27,84,40,95)(28,85,41,96)(29,86,42,97)(30,87,43,98)(31,88,44,99)(32,89,45,100)(33,90,46,101)(34,91,47,102)(35,79,48,103)(36,80,49,104)(37,81,50,92)(38,82,51,93)(39,83,52,94)>;
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), (1,65)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,91)(15,79)(16,80)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(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), (1,35)(2,36)(3,37)(4,38)(5,39)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,102)(15,103)(16,104)(17,92)(18,93)(19,94)(20,95)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,65)(49,53)(50,54)(51,55)(52,56)(66,91)(67,79)(68,80)(69,81)(70,82)(71,83)(72,84)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90), (1,67,65,15)(2,68,53,16)(3,69,54,17)(4,70,55,18)(5,71,56,19)(6,72,57,20)(7,73,58,21)(8,74,59,22)(9,75,60,23)(10,76,61,24)(11,77,62,25)(12,78,63,26)(13,66,64,14)(27,84,40,95)(28,85,41,96)(29,86,42,97)(30,87,43,98)(31,88,44,99)(32,89,45,100)(33,90,46,101)(34,91,47,102)(35,79,48,103)(36,80,49,104)(37,81,50,92)(38,82,51,93)(39,83,52,94) );
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)], [(1,65),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,91),(15,79),(16,80),(17,81),(18,82),(19,83),(20,84),(21,85),(22,86),(23,87),(24,88),(25,89),(26,90),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52),(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)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,102),(15,103),(16,104),(17,92),(18,93),(19,94),(20,95),(21,96),(22,97),(23,98),(24,99),(25,100),(26,101),(40,57),(41,58),(42,59),(43,60),(44,61),(45,62),(46,63),(47,64),(48,65),(49,53),(50,54),(51,55),(52,56),(66,91),(67,79),(68,80),(69,81),(70,82),(71,83),(72,84),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90)], [(1,67,65,15),(2,68,53,16),(3,69,54,17),(4,70,55,18),(5,71,56,19),(6,72,57,20),(7,73,58,21),(8,74,59,22),(9,75,60,23),(10,76,61,24),(11,77,62,25),(12,78,63,26),(13,66,64,14),(27,84,40,95),(28,85,41,96),(29,86,42,97),(30,87,43,98),(31,88,44,99),(32,89,45,100),(33,90,46,101),(34,91,47,102),(35,79,48,103),(36,80,49,104),(37,81,50,92),(38,82,51,93),(39,83,52,94)]])
C13×C22⋊C4 is a maximal subgroup of
C22.2D52 C23.11D26 C22⋊Dic26 C23.D26 Dic13⋊4D4 C22⋊D52 D26.12D4 D26⋊D4 C23.6D26 C22.D52 D4×C52
130 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 13A | ··· | 13L | 26A | ··· | 26AJ | 26AK | ··· | 26BH | 52A | ··· | 52AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C13 | C26 | C26 | C52 | D4 | D4×C13 |
| kernel | C13×C22⋊C4 | C2×C52 | C22×C26 | C2×C26 | C22⋊C4 | C2×C4 | C23 | C22 | C26 | C2 |
| # reps | 1 | 2 | 1 | 4 | 12 | 24 | 12 | 48 | 2 | 24 |
Matrix representation of C13×C22⋊C4 ►in GL3(𝔽53) generated by
| 1 | 0 | 0 |
| 0 | 24 | 0 |
| 0 | 0 | 24 |
| 1 | 0 | 0 |
| 0 | 52 | 0 |
| 0 | 11 | 1 |
| 1 | 0 | 0 |
| 0 | 52 | 0 |
| 0 | 0 | 52 |
| 30 | 0 | 0 |
| 0 | 42 | 51 |
| 0 | 8 | 11 |
G:=sub<GL(3,GF(53))| [1,0,0,0,24,0,0,0,24],[1,0,0,0,52,11,0,0,1],[1,0,0,0,52,0,0,0,52],[30,0,0,0,42,8,0,51,11] >;
C13×C22⋊C4 in GAP, Magma, Sage, TeX
C_{13}\times C_2^2\rtimes C_4 % in TeX
G:=Group("C13xC2^2:C4"); // GroupNames label
G:=SmallGroup(208,21);
// by ID
G=gap.SmallGroup(208,21);
# by ID
G:=PCGroup([5,-2,-2,-13,-2,-2,520,541]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
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