direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic13, C26⋊2C4, C2.2D26, C22.D13, C26.4C22, C13⋊3(C2×C4), (C2×C26).C2, SmallGroup(104,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C2×Dic13 |
Generators and relations for C2×Dic13
G = < a,b,c | a2=b26=1, c2=b13, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic13
►Regular action on 104 points
Generators in S104
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(53 92)(54 93)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(61 100)(62 101)(63 102)(64 103)(65 104)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)
(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 53 14 66)(2 78 15 65)(3 77 16 64)(4 76 17 63)(5 75 18 62)(6 74 19 61)(7 73 20 60)(8 72 21 59)(9 71 22 58)(10 70 23 57)(11 69 24 56)(12 68 25 55)(13 67 26 54)(27 104 40 91)(28 103 41 90)(29 102 42 89)(30 101 43 88)(31 100 44 87)(32 99 45 86)(33 98 46 85)(34 97 47 84)(35 96 48 83)(36 95 49 82)(37 94 50 81)(38 93 51 80)(39 92 52 79)
G:=sub<Sym(104)| (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91), (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), (1,53,14,66)(2,78,15,65)(3,77,16,64)(4,76,17,63)(5,75,18,62)(6,74,19,61)(7,73,20,60)(8,72,21,59)(9,71,22,58)(10,70,23,57)(11,69,24,56)(12,68,25,55)(13,67,26,54)(27,104,40,91)(28,103,41,90)(29,102,42,89)(30,101,43,88)(31,100,44,87)(32,99,45,86)(33,98,46,85)(34,97,47,84)(35,96,48,83)(36,95,49,82)(37,94,50,81)(38,93,51,80)(39,92,52,79)>;
G:=Group( (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91), (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), (1,53,14,66)(2,78,15,65)(3,77,16,64)(4,76,17,63)(5,75,18,62)(6,74,19,61)(7,73,20,60)(8,72,21,59)(9,71,22,58)(10,70,23,57)(11,69,24,56)(12,68,25,55)(13,67,26,54)(27,104,40,91)(28,103,41,90)(29,102,42,89)(30,101,43,88)(31,100,44,87)(32,99,45,86)(33,98,46,85)(34,97,47,84)(35,96,48,83)(36,95,49,82)(37,94,50,81)(38,93,51,80)(39,92,52,79) );
G=PermutationGroup([[(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(53,92),(54,93),(55,94),(56,95),(57,96),(58,97),(59,98),(60,99),(61,100),(62,101),(63,102),(64,103),(65,104),(66,79),(67,80),(68,81),(69,82),(70,83),(71,84),(72,85),(73,86),(74,87),(75,88),(76,89),(77,90),(78,91)], [(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)], [(1,53,14,66),(2,78,15,65),(3,77,16,64),(4,76,17,63),(5,75,18,62),(6,74,19,61),(7,73,20,60),(8,72,21,59),(9,71,22,58),(10,70,23,57),(11,69,24,56),(12,68,25,55),(13,67,26,54),(27,104,40,91),(28,103,41,90),(29,102,42,89),(30,101,43,88),(31,100,44,87),(32,99,45,86),(33,98,46,85),(34,97,47,84),(35,96,48,83),(36,95,49,82),(37,94,50,81),(38,93,51,80),(39,92,52,79)]])
C2×Dic13 is a maximal subgroup of
C26.D4 C52⋊3C4 D26⋊C4 C23.D13 C13⋊M4(2) C2×C4×D13 D4⋊2D13
C2×Dic13 is a maximal quotient of C52.4C4 C52⋊3C4 C23.D13
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 13A | ··· | 13F | 26A | ··· | 26R |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 |
| size | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 2 | ··· | 2 | 2 | ··· | 2 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | D13 | Dic13 | D26 |
| kernel | C2×Dic13 | Dic13 | C2×C26 | C26 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 |
Matrix representation of C2×Dic13 ►in GL3(𝔽53) generated by
| 52 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 52 |
| 0 | 1 | 27 |
| 1 | 0 | 0 |
| 0 | 22 | 35 |
| 0 | 24 | 31 |
G:=sub<GL(3,GF(53))| [52,0,0,0,1,0,0,0,1],[1,0,0,0,0,1,0,52,27],[1,0,0,0,22,24,0,35,31] >;
C2×Dic13 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{13} % in TeX
G:=Group("C2xDic13"); // GroupNames label
G:=SmallGroup(104,7);
// by ID
G=gap.SmallGroup(104,7);
# by ID
G:=PCGroup([4,-2,-2,-2,-13,16,1539]);
// Polycyclic
G:=Group<a,b,c|a^2=b^26=1,c^2=b^13,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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