direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C23×D13, C13⋊C24, C26⋊C23, (C22×C26)⋊3C2, (C2×C26)⋊4C22, SmallGroup(208,50)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C23×D13 |
Generators and relations for C23×D13
G = < a,b,c,d,e | a2=b2=c2=d13=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 746 in 134 conjugacy classes, 83 normal (5 characteristic)
C1, C2, C2, C22, C22, C23, C23, C13, C24, D13, C26, D26, C2×C26, C22×D13, C22×C26, C23×D13
Quotients: C1, C2, C22, C23, C24, D13, D26, C22×D13, C23×D13
►On 104 points
Generators in S104
(1 60)(2 61)(3 62)(4 63)(5 64)(6 65)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 79)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 92)(51 93)(52 94)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 27)(53 101)(54 102)(55 103)(56 104)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 99)(65 100)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(76 91)(77 79)(78 80)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 14)(10 15)(11 16)(12 17)(13 18)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 40)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(53 77)(54 78)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)(61 72)(62 73)(63 74)(64 75)(65 76)(79 101)(80 102)(81 103)(82 104)(83 92)(84 93)(85 94)(86 95)(87 96)(88 97)(89 98)(90 99)(91 100)
(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 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 65)(9 64)(10 63)(11 62)(12 61)(13 60)(14 75)(15 74)(16 73)(17 72)(18 71)(19 70)(20 69)(21 68)(22 67)(23 66)(24 78)(25 77)(26 76)(27 91)(28 90)(29 89)(30 88)(31 87)(32 86)(33 85)(34 84)(35 83)(36 82)(37 81)(38 80)(39 79)(40 94)(41 93)(42 92)(43 104)(44 103)(45 102)(46 101)(47 100)(48 99)(49 98)(50 97)(51 96)(52 95)
G:=sub<Sym(104)| (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,79)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,92)(51,93)(52,94), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,27)(53,101)(54,102)(55,103)(56,104)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,99)(65,100)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,79)(78,80), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,14)(10,15)(11,16)(12,17)(13,18)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,40)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(53,77)(54,78)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(79,101)(80,102)(81,103)(82,104)(83,92)(84,93)(85,94)(86,95)(87,96)(88,97)(89,98)(90,99)(91,100), (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), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,65)(9,64)(10,63)(11,62)(12,61)(13,60)(14,75)(15,74)(16,73)(17,72)(18,71)(19,70)(20,69)(21,68)(22,67)(23,66)(24,78)(25,77)(26,76)(27,91)(28,90)(29,89)(30,88)(31,87)(32,86)(33,85)(34,84)(35,83)(36,82)(37,81)(38,80)(39,79)(40,94)(41,93)(42,92)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)>;
G:=Group( (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,79)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,92)(51,93)(52,94), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,27)(53,101)(54,102)(55,103)(56,104)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,99)(65,100)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,79)(78,80), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,14)(10,15)(11,16)(12,17)(13,18)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,40)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(53,77)(54,78)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(79,101)(80,102)(81,103)(82,104)(83,92)(84,93)(85,94)(86,95)(87,96)(88,97)(89,98)(90,99)(91,100), (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), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,65)(9,64)(10,63)(11,62)(12,61)(13,60)(14,75)(15,74)(16,73)(17,72)(18,71)(19,70)(20,69)(21,68)(22,67)(23,66)(24,78)(25,77)(26,76)(27,91)(28,90)(29,89)(30,88)(31,87)(32,86)(33,85)(34,84)(35,83)(36,82)(37,81)(38,80)(39,79)(40,94)(41,93)(42,92)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95) );
G=PermutationGroup([[(1,60),(2,61),(3,62),(4,63),(5,64),(6,65),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,59),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,72),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,80),(28,81),(29,82),(30,83),(31,84),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,91),(39,79),(40,95),(41,96),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,92),(51,93),(52,94)], [(1,40),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,28),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,39),(26,27),(53,101),(54,102),(55,103),(56,104),(57,92),(58,93),(59,94),(60,95),(61,96),(62,97),(63,98),(64,99),(65,100),(66,81),(67,82),(68,83),(69,84),(70,85),(71,86),(72,87),(73,88),(74,89),(75,90),(76,91),(77,79),(78,80)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,14),(10,15),(11,16),(12,17),(13,18),(27,47),(28,48),(29,49),(30,50),(31,51),(32,52),(33,40),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(53,77),(54,78),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71),(61,72),(62,73),(63,74),(64,75),(65,76),(79,101),(80,102),(81,103),(82,104),(83,92),(84,93),(85,94),(86,95),(87,96),(88,97),(89,98),(90,99),(91,100)], [(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)], [(1,59),(2,58),(3,57),(4,56),(5,55),(6,54),(7,53),(8,65),(9,64),(10,63),(11,62),(12,61),(13,60),(14,75),(15,74),(16,73),(17,72),(18,71),(19,70),(20,69),(21,68),(22,67),(23,66),(24,78),(25,77),(26,76),(27,91),(28,90),(29,89),(30,88),(31,87),(32,86),(33,85),(34,84),(35,83),(36,82),(37,81),(38,80),(39,79),(40,94),(41,93),(42,92),(43,104),(44,103),(45,102),(46,101),(47,100),(48,99),(49,98),(50,97),(51,96),(52,95)]])
C23×D13 is a maximal subgroup of
C22⋊D52 C23⋊D26
C23×D13 is a maximal quotient of D4⋊6D26 Q8.10D26 D4⋊8D26 D4.10D26
64 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 13A | ··· | 13F | 26A | ··· | 26AP |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 13 | ··· | 13 | 26 | ··· | 26 |
| size | 1 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D13 | D26 |
| kernel | C23×D13 | C22×D13 | C22×C26 | C23 | C22 |
| # reps | 1 | 14 | 1 | 6 | 42 |
Matrix representation of C23×D13 ►in GL4(𝔽53) generated by
| 1 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 52 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 52 | 13 |
| 52 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 52 |
| 0 | 0 | 52 | 0 |
G:=sub<GL(4,GF(53))| [1,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[1,0,0,0,0,1,0,0,0,0,0,52,0,0,1,13],[52,0,0,0,0,1,0,0,0,0,0,52,0,0,52,0] >;
C23×D13 in GAP, Magma, Sage, TeX
C_2^3\times D_{13} % in TeX
G:=Group("C2^3xD13"); // GroupNames label
G:=SmallGroup(208,50);
// by ID
G=gap.SmallGroup(208,50);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,4804]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^13=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations