direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C26.C6, C26⋊2C12, Dic13⋊3C6, (C2×C26).C6, C13⋊3(C2×C12), (C2×Dic13)⋊C3, C26.4(C2×C6), C22.(C13⋊C6), C13⋊C3⋊3(C2×C4), (C2×C13⋊C3)⋊2C4, C2.2(C2×C13⋊C6), (C22×C13⋊C3).C2, (C2×C13⋊C3).4C22, SmallGroup(312,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C2×C26.C6 |
Generators and relations for C2×C26.C6
G = < a,b,c | a2=b26=1, c6=b13, ab=ba, ac=ca, cbc-1=b23 >
Smallest permutation representation of C2×C26.C6
►On 104 points
Generators in S104
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(53 99)(54 100)(55 101)(56 102)(57 103)(58 104)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)(73 93)(74 94)(75 95)(76 96)(77 97)(78 98)
(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 14 71)(2 75 17 70 10 55 15 62 4 57 23 68)(3 66 20 69 19 78 16 53 7 56 6 65)(5 74 26 67 11 72 18 61 13 54 24 59)(8 73 9 64 12 63 21 60 22 77 25 76)(27 101 32 82 47 103 40 88 45 95 34 90)(28 92 35 81 30 100 41 79 48 94 43 87)(29 83 38 80 39 97 42 96 51 93 52 84)(31 91 44 104)(33 99 50 102 49 85 46 86 37 89 36 98)
G:=sub<Sym(104)| (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(53,99)(54,100)(55,101)(56,102)(57,103)(58,104)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98), (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), (1,58,14,71)(2,75,17,70,10,55,15,62,4,57,23,68)(3,66,20,69,19,78,16,53,7,56,6,65)(5,74,26,67,11,72,18,61,13,54,24,59)(8,73,9,64,12,63,21,60,22,77,25,76)(27,101,32,82,47,103,40,88,45,95,34,90)(28,92,35,81,30,100,41,79,48,94,43,87)(29,83,38,80,39,97,42,96,51,93,52,84)(31,91,44,104)(33,99,50,102,49,85,46,86,37,89,36,98)>;
G:=Group( (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(53,99)(54,100)(55,101)(56,102)(57,103)(58,104)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98), (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), (1,58,14,71)(2,75,17,70,10,55,15,62,4,57,23,68)(3,66,20,69,19,78,16,53,7,56,6,65)(5,74,26,67,11,72,18,61,13,54,24,59)(8,73,9,64,12,63,21,60,22,77,25,76)(27,101,32,82,47,103,40,88,45,95,34,90)(28,92,35,81,30,100,41,79,48,94,43,87)(29,83,38,80,39,97,42,96,51,93,52,84)(31,91,44,104)(33,99,50,102,49,85,46,86,37,89,36,98) );
G=PermutationGroup([[(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(53,99),(54,100),(55,101),(56,102),(57,103),(58,104),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,91),(72,92),(73,93),(74,94),(75,95),(76,96),(77,97),(78,98)], [(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)], [(1,58,14,71),(2,75,17,70,10,55,15,62,4,57,23,68),(3,66,20,69,19,78,16,53,7,56,6,65),(5,74,26,67,11,72,18,61,13,54,24,59),(8,73,9,64,12,63,21,60,22,77,25,76),(27,101,32,82,47,103,40,88,45,95,34,90),(28,92,35,81,30,100,41,79,48,94,43,87),(29,83,38,80,39,97,42,96,51,93,52,84),(31,91,44,104),(33,99,50,102,49,85,46,86,37,89,36,98)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 12A | ··· | 12H | 13A | 13B | 26A | ··· | 26F |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 13 | 13 | 26 | ··· | 26 |
| size | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | ··· | 13 | 13 | ··· | 13 | 6 | 6 | 6 | ··· | 6 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | C13⋊C6 | C26.C6 | C2×C13⋊C6 |
| kernel | C2×C26.C6 | C26.C6 | C22×C13⋊C3 | C2×Dic13 | C2×C13⋊C3 | Dic13 | C2×C26 | C26 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 2 | 4 | 2 |
Matrix representation of C2×C26.C6 ►in GL8(𝔽157)
| 156 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 156 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 156 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 156 | 67 | 155 | 68 | 155 | 67 |
| 0 | 0 | 90 | 92 | 90 | 1 | 91 | 91 |
| 0 | 0 | 66 | 64 | 67 | 155 | 133 | 65 |
| 0 | 0 | 92 | 25 | 91 | 91 | 25 | 92 |
| 0 | 0 | 65 | 133 | 155 | 67 | 64 | 66 |
| 0 | 0 | 91 | 91 | 1 | 90 | 92 | 90 |
| 135 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 144 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 64 | 67 | 154 | 134 | 154 |
| 0 | 0 | 8 | 17 | 38 | 60 | 62 | 114 |
| 0 | 0 | 120 | 85 | 30 | 106 | 69 | 136 |
| 0 | 0 | 67 | 154 | 137 | 87 | 83 | 154 |
| 0 | 0 | 3 | 126 | 70 | 20 | 3 | 90 |
| 0 | 0 | 43 | 110 | 103 | 97 | 146 | 7 |
G:=sub<GL(8,GF(157))| [156,0,0,0,0,0,0,0,0,156,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[156,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,156,90,66,92,65,91,0,0,67,92,64,25,133,91,0,0,155,90,67,91,155,1,0,0,68,1,155,91,67,90,0,0,155,91,133,25,64,92,0,0,67,91,65,92,66,90],[135,0,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,0,13,8,120,67,3,43,0,0,64,17,85,154,126,110,0,0,67,38,30,137,70,103,0,0,154,60,106,87,20,97,0,0,134,62,69,83,3,146,0,0,154,114,136,154,90,7] >;
C2×C26.C6 in GAP, Magma, Sage, TeX
C_2\times C_{26}.C_6 % in TeX
G:=Group("C2xC26.C6"); // GroupNames label
G:=SmallGroup(312,11);
// by ID
G=gap.SmallGroup(312,11);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-13,60,7204,464]);
// Polycyclic
G:=Group<a,b,c|a^2=b^26=1,c^6=b^13,a*b=b*a,a*c=c*a,c*b*c^-1=b^23>;
// generators/relations
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