metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23⋊Dic13, C23.2D26, (C2×C52)⋊6C4, (C2×C4)⋊Dic13, (C2×C26).2D4, C13⋊4(C23⋊C4), (C22×C26)⋊2C4, (D4×C26).6C2, (C2×D4).3D13, C23.D13⋊2C2, C26.26(C22⋊C4), C22.2(C13⋊D4), (C22×C26).6C22, C2.5(C23.D13), C22.3(C2×Dic13), (C2×C26).49(C2×C4), SmallGroup(416,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23⋊Dic13
G = < a,b,c,d,e | a2=b2=c2=d26=1, e2=d13, ab=ba, dad-1=ac=ca, eae-1=abc, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >
2C2
2C2
2C2
4C2
2C22
2C4
4C22
4C22
52C4
52C4
2C26
2C26
2C26
4C26
2D4
2D4
26C2×C4
26C2×C4
2C52
13C22⋊C4
13C22⋊C4
13C23⋊C4
Smallest permutation representation of C23⋊Dic13
►On 104 points
Generators in S104
(1 74)(2 43)(3 76)(4 45)(5 78)(6 47)(7 54)(8 49)(9 56)(10 51)(11 58)(12 27)(13 60)(14 29)(15 62)(16 31)(17 64)(18 33)(19 66)(20 35)(21 68)(22 37)(23 70)(24 39)(25 72)(26 41)(28 83)(30 85)(32 87)(34 89)(36 91)(38 93)(40 95)(42 97)(44 99)(46 101)(48 103)(50 79)(52 81)(53 102)(55 104)(57 80)(59 82)(61 84)(63 86)(65 88)(67 90)(69 92)(71 94)(73 96)(75 98)(77 100)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 79)(10 80)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 53)(48 54)(49 55)(50 56)(51 57)(52 58)
(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 74 14 61)(2 73 15 60)(3 72 16 59)(4 71 17 58)(5 70 18 57)(6 69 19 56)(7 68 20 55)(8 67 21 54)(9 66 22 53)(10 65 23 78)(11 64 24 77)(12 63 25 76)(13 62 26 75)(27 99 40 86)(28 98 41 85)(29 97 42 84)(30 96 43 83)(31 95 44 82)(32 94 45 81)(33 93 46 80)(34 92 47 79)(35 91 48 104)(36 90 49 103)(37 89 50 102)(38 88 51 101)(39 87 52 100)
G:=sub<Sym(104)| (1,74)(2,43)(3,76)(4,45)(5,78)(6,47)(7,54)(8,49)(9,56)(10,51)(11,58)(12,27)(13,60)(14,29)(15,62)(16,31)(17,64)(18,33)(19,66)(20,35)(21,68)(22,37)(23,70)(24,39)(25,72)(26,41)(28,83)(30,85)(32,87)(34,89)(36,91)(38,93)(40,95)(42,97)(44,99)(46,101)(48,103)(50,79)(52,81)(53,102)(55,104)(57,80)(59,82)(61,84)(63,86)(65,88)(67,90)(69,92)(71,94)(73,96)(75,98)(77,100), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,53)(48,54)(49,55)(50,56)(51,57)(52,58), (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), (1,74,14,61)(2,73,15,60)(3,72,16,59)(4,71,17,58)(5,70,18,57)(6,69,19,56)(7,68,20,55)(8,67,21,54)(9,66,22,53)(10,65,23,78)(11,64,24,77)(12,63,25,76)(13,62,26,75)(27,99,40,86)(28,98,41,85)(29,97,42,84)(30,96,43,83)(31,95,44,82)(32,94,45,81)(33,93,46,80)(34,92,47,79)(35,91,48,104)(36,90,49,103)(37,89,50,102)(38,88,51,101)(39,87,52,100)>;
G:=Group( (1,74)(2,43)(3,76)(4,45)(5,78)(6,47)(7,54)(8,49)(9,56)(10,51)(11,58)(12,27)(13,60)(14,29)(15,62)(16,31)(17,64)(18,33)(19,66)(20,35)(21,68)(22,37)(23,70)(24,39)(25,72)(26,41)(28,83)(30,85)(32,87)(34,89)(36,91)(38,93)(40,95)(42,97)(44,99)(46,101)(48,103)(50,79)(52,81)(53,102)(55,104)(57,80)(59,82)(61,84)(63,86)(65,88)(67,90)(69,92)(71,94)(73,96)(75,98)(77,100), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,53)(48,54)(49,55)(50,56)(51,57)(52,58), (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), (1,74,14,61)(2,73,15,60)(3,72,16,59)(4,71,17,58)(5,70,18,57)(6,69,19,56)(7,68,20,55)(8,67,21,54)(9,66,22,53)(10,65,23,78)(11,64,24,77)(12,63,25,76)(13,62,26,75)(27,99,40,86)(28,98,41,85)(29,97,42,84)(30,96,43,83)(31,95,44,82)(32,94,45,81)(33,93,46,80)(34,92,47,79)(35,91,48,104)(36,90,49,103)(37,89,50,102)(38,88,51,101)(39,87,52,100) );
G=PermutationGroup([[(1,74),(2,43),(3,76),(4,45),(5,78),(6,47),(7,54),(8,49),(9,56),(10,51),(11,58),(12,27),(13,60),(14,29),(15,62),(16,31),(17,64),(18,33),(19,66),(20,35),(21,68),(22,37),(23,70),(24,39),(25,72),(26,41),(28,83),(30,85),(32,87),(34,89),(36,91),(38,93),(40,95),(42,97),(44,99),(46,101),(48,103),(50,79),(52,81),(53,102),(55,104),(57,80),(59,82),(61,84),(63,86),(65,88),(67,90),(69,92),(71,94),(73,96),(75,98),(77,100)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(27,72),(28,73),(29,74),(30,75),(31,76),(32,77),(33,78),(34,53),(35,54),(36,55),(37,56),(38,57),(39,58),(40,59),(41,60),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67),(49,68),(50,69),(51,70),(52,71),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97),(85,98),(86,99),(87,100),(88,101),(89,102),(90,103),(91,104)], [(1,97),(2,98),(3,99),(4,100),(5,101),(6,102),(7,103),(8,104),(9,79),(10,80),(11,81),(12,82),(13,83),(14,84),(15,85),(16,86),(17,87),(18,88),(19,89),(20,90),(21,91),(22,92),(23,93),(24,94),(25,95),(26,96),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72),(41,73),(42,74),(43,75),(44,76),(45,77),(46,78),(47,53),(48,54),(49,55),(50,56),(51,57),(52,58)], [(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)], [(1,74,14,61),(2,73,15,60),(3,72,16,59),(4,71,17,58),(5,70,18,57),(6,69,19,56),(7,68,20,55),(8,67,21,54),(9,66,22,53),(10,65,23,78),(11,64,24,77),(12,63,25,76),(13,62,26,75),(27,99,40,86),(28,98,41,85),(29,97,42,84),(30,96,43,83),(31,95,44,82),(32,94,45,81),(33,93,46,80),(34,92,47,79),(35,91,48,104),(36,90,49,103),(37,89,50,102),(38,88,51,101),(39,87,52,100)]])
71 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 13A | ··· | 13F | 26A | ··· | 26R | 26S | ··· | 26AP | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 52 | 52 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
71 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | D13 | Dic13 | Dic13 | D26 | C13⋊D4 | C23⋊C4 | C23⋊Dic13 |
| kernel | C23⋊Dic13 | C23.D13 | D4×C26 | C2×C52 | C22×C26 | C2×C26 | C2×D4 | C2×C4 | C23 | C23 | C22 | C13 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 24 | 1 | 12 |
Matrix representation of C23⋊Dic13 ►in GL4(𝔽53) generated by
| 6 | 1 | 0 | 0 |
| 18 | 47 | 0 | 0 |
| 32 | 24 | 42 | 42 |
| 44 | 14 | 35 | 11 |
| 6 | 1 | 0 | 0 |
| 18 | 47 | 0 | 0 |
| 0 | 0 | 11 | 11 |
| 0 | 0 | 18 | 42 |
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 22 | 13 | 41 | 21 |
| 22 | 25 | 0 | 41 |
| 0 | 0 | 19 | 16 |
| 0 | 0 | 31 | 40 |
| 37 | 14 | 15 | 43 |
| 46 | 16 | 31 | 50 |
| 22 | 1 | 41 | 27 |
| 27 | 12 | 28 | 12 |
G:=sub<GL(4,GF(53))| [6,18,32,44,1,47,24,14,0,0,42,35,0,0,42,11],[6,18,0,0,1,47,0,0,0,0,11,18,0,0,11,42],[52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[22,22,0,0,13,25,0,0,41,0,19,31,21,41,16,40],[37,46,22,27,14,16,1,12,15,31,41,28,43,50,27,12] >;
C23⋊Dic13 in GAP, Magma, Sage, TeX
C_2^3\rtimes {\rm Dic}_{13} % in TeX
G:=Group("C2^3:Dic13"); // GroupNames label
G:=SmallGroup(416,41);
// by ID
G=gap.SmallGroup(416,41);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,188,579,13829]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^26=1,e^2=d^13,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e^-1=a*b*c,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
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