direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×2+ 1+4, C26.19C24, C52.51C23, C4○D4⋊3C26, D4⋊4(C2×C26), (C2×D4)⋊6C26, Q8⋊4(C2×C26), (D4×C26)⋊15C2, (C2×C52)⋊9C22, C23⋊2(C2×C26), C4.9(C22×C26), C2.4(C23×C26), (C2×C26).7C23, (D4×C13)⋊13C22, (C22×C26)⋊2C22, (Q8×C13)⋊12C22, C22.2(C22×C26), (C2×C4)⋊2(C2×C26), (C13×C4○D4)⋊8C2, SmallGroup(416,231)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×2+ 1+4
G = < a,b,c,d,e | a13=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
Subgroups: 220 in 166 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, D4, Q8, C23, C13, C2×D4, C4○D4, C26, C26, 2+ 1+4, C52, C2×C26, C2×C26, C2×C52, D4×C13, Q8×C13, C22×C26, D4×C26, C13×C4○D4, C13×2+ 1+4
Quotients: C1, C2, C22, C23, C13, C24, C26, 2+ 1+4, C2×C26, C22×C26, C23×C26, C13×2+ 1+4
►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)
(1 100 19 43)(2 101 20 44)(3 102 21 45)(4 103 22 46)(5 104 23 47)(6 92 24 48)(7 93 25 49)(8 94 26 50)(9 95 14 51)(10 96 15 52)(11 97 16 40)(12 98 17 41)(13 99 18 42)(27 56 79 71)(28 57 80 72)(29 58 81 73)(30 59 82 74)(31 60 83 75)(32 61 84 76)(33 62 85 77)(34 63 86 78)(35 64 87 66)(36 65 88 67)(37 53 89 68)(38 54 90 69)(39 55 91 70)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 53)(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)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 92)(88 93)(89 94)(90 95)(91 96)
(1 43 19 100)(2 44 20 101)(3 45 21 102)(4 46 22 103)(5 47 23 104)(6 48 24 92)(7 49 25 93)(8 50 26 94)(9 51 14 95)(10 52 15 96)(11 40 16 97)(12 41 17 98)(13 42 18 99)(27 56 79 71)(28 57 80 72)(29 58 81 73)(30 59 82 74)(31 60 83 75)(32 61 84 76)(33 62 85 77)(34 63 86 78)(35 64 87 66)(36 65 88 67)(37 53 89 68)(38 54 90 69)(39 55 91 70)
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 79)(12 80)(13 81)(14 38)(15 39)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(23 34)(24 35)(25 36)(26 37)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)(49 65)(50 53)(51 54)(52 55)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)
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), (1,100,19,43)(2,101,20,44)(3,102,21,45)(4,103,22,46)(5,104,23,47)(6,92,24,48)(7,93,25,49)(8,94,26,50)(9,95,14,51)(10,96,15,52)(11,97,16,40)(12,98,17,41)(13,99,18,42)(27,56,79,71)(28,57,80,72)(29,58,81,73)(30,59,82,74)(31,60,83,75)(32,61,84,76)(33,62,85,77)(34,63,86,78)(35,64,87,66)(36,65,88,67)(37,53,89,68)(38,54,90,69)(39,55,91,70), (1,74)(2,75)(3,76)(4,77)(5,78)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,53)(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)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,92)(88,93)(89,94)(90,95)(91,96), (1,43,19,100)(2,44,20,101)(3,45,21,102)(4,46,22,103)(5,47,23,104)(6,48,24,92)(7,49,25,93)(8,50,26,94)(9,51,14,95)(10,52,15,96)(11,40,16,97)(12,41,17,98)(13,42,18,99)(27,56,79,71)(28,57,80,72)(29,58,81,73)(30,59,82,74)(31,60,83,75)(32,61,84,76)(33,62,85,77)(34,63,86,78)(35,64,87,66)(36,65,88,67)(37,53,89,68)(38,54,90,69)(39,55,91,70), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,79)(12,80)(13,81)(14,38)(15,39)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35)(25,36)(26,37)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(49,65)(50,53)(51,54)(52,55)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)>;
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), (1,100,19,43)(2,101,20,44)(3,102,21,45)(4,103,22,46)(5,104,23,47)(6,92,24,48)(7,93,25,49)(8,94,26,50)(9,95,14,51)(10,96,15,52)(11,97,16,40)(12,98,17,41)(13,99,18,42)(27,56,79,71)(28,57,80,72)(29,58,81,73)(30,59,82,74)(31,60,83,75)(32,61,84,76)(33,62,85,77)(34,63,86,78)(35,64,87,66)(36,65,88,67)(37,53,89,68)(38,54,90,69)(39,55,91,70), (1,74)(2,75)(3,76)(4,77)(5,78)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,53)(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)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,92)(88,93)(89,94)(90,95)(91,96), (1,43,19,100)(2,44,20,101)(3,45,21,102)(4,46,22,103)(5,47,23,104)(6,48,24,92)(7,49,25,93)(8,50,26,94)(9,51,14,95)(10,52,15,96)(11,40,16,97)(12,41,17,98)(13,42,18,99)(27,56,79,71)(28,57,80,72)(29,58,81,73)(30,59,82,74)(31,60,83,75)(32,61,84,76)(33,62,85,77)(34,63,86,78)(35,64,87,66)(36,65,88,67)(37,53,89,68)(38,54,90,69)(39,55,91,70), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,79)(12,80)(13,81)(14,38)(15,39)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35)(25,36)(26,37)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(49,65)(50,53)(51,54)(52,55)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104) );
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)], [(1,100,19,43),(2,101,20,44),(3,102,21,45),(4,103,22,46),(5,104,23,47),(6,92,24,48),(7,93,25,49),(8,94,26,50),(9,95,14,51),(10,96,15,52),(11,97,16,40),(12,98,17,41),(13,99,18,42),(27,56,79,71),(28,57,80,72),(29,58,81,73),(30,59,82,74),(31,60,83,75),(32,61,84,76),(33,62,85,77),(34,63,86,78),(35,64,87,66),(36,65,88,67),(37,53,89,68),(38,54,90,69),(39,55,91,70)], [(1,74),(2,75),(3,76),(4,77),(5,78),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,73),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,53),(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),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,103),(86,104),(87,92),(88,93),(89,94),(90,95),(91,96)], [(1,43,19,100),(2,44,20,101),(3,45,21,102),(4,46,22,103),(5,47,23,104),(6,48,24,92),(7,49,25,93),(8,50,26,94),(9,51,14,95),(10,52,15,96),(11,40,16,97),(12,41,17,98),(13,42,18,99),(27,56,79,71),(28,57,80,72),(29,58,81,73),(30,59,82,74),(31,60,83,75),(32,61,84,76),(33,62,85,77),(34,63,86,78),(35,64,87,66),(36,65,88,67),(37,53,89,68),(38,54,90,69),(39,55,91,70)], [(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,91),(11,79),(12,80),(13,81),(14,38),(15,39),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33),(23,34),(24,35),(25,36),(26,37),(40,56),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64),(49,65),(50,53),(51,54),(52,55),(66,92),(67,93),(68,94),(69,95),(70,96),(71,97),(72,98),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104)]])
221 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2J | 4A | ··· | 4F | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26DP | 52A | ··· | 52BT |
| order | 1 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
221 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C13 | C26 | C26 | 2+ 1+4 | C13×2+ 1+4 |
| kernel | C13×2+ 1+4 | D4×C26 | C13×C4○D4 | 2+ 1+4 | C2×D4 | C4○D4 | C13 | C1 |
| # reps | 1 | 9 | 6 | 12 | 108 | 72 | 1 | 12 |
Matrix representation of C13×2+ 1+4 ►in GL4(𝔽53) generated by
| 24 | 0 | 0 | 0 |
| 0 | 24 | 0 | 0 |
| 0 | 0 | 24 | 0 |
| 0 | 0 | 0 | 24 |
| 1 | 8 | 0 | 2 |
| 0 | 7 | 51 | 0 |
| 0 | 25 | 46 | 0 |
| 52 | 21 | 8 | 52 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 4 | 4 | 0 | 1 |
| 49 | 49 | 1 | 0 |
| 52 | 45 | 0 | 51 |
| 0 | 7 | 51 | 0 |
| 0 | 25 | 46 | 0 |
| 1 | 29 | 8 | 1 |
| 0 | 7 | 51 | 0 |
| 52 | 45 | 0 | 51 |
| 49 | 25 | 0 | 46 |
| 4 | 28 | 1 | 8 |
G:=sub<GL(4,GF(53))| [24,0,0,0,0,24,0,0,0,0,24,0,0,0,0,24],[1,0,0,52,8,7,25,21,0,51,46,8,2,0,0,52],[0,1,4,49,1,0,4,49,0,0,0,1,0,0,1,0],[52,0,0,1,45,7,25,29,0,51,46,8,51,0,0,1],[0,52,49,4,7,45,25,28,51,0,0,1,0,51,46,8] >;
C13×2+ 1+4 in GAP, Magma, Sage, TeX
C_{13}\times 2_+^{1+4} % in TeX
G:=Group("C13xES+(2,2)"); // GroupNames label
G:=SmallGroup(416,231);
// by ID
G=gap.SmallGroup(416,231);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-13,-2,2521,1916,5187]);
// Polycyclic
G:=Group<a,b,c,d,e|a^13=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations