direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C13⋊D4, C26⋊2D4, C23⋊D13, C22⋊2D26, D26⋊3C22, C26.10C23, Dic13⋊2C22, C13⋊3(C2×D4), (C2×C26)⋊3C22, (C22×C26)⋊2C2, (C2×Dic13)⋊4C2, (C22×D13)⋊3C2, C2.10(C22×D13), SmallGroup(208,44)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C13⋊D4
G = < a,b,c,d | a2=b13=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 298 in 54 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C23, C13, C2×D4, D13, C26, C26, C26, Dic13, D26, D26, C2×C26, C2×C26, C2×C26, C2×Dic13, C13⋊D4, C22×D13, C22×C26, C2×C13⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, D26, C13⋊D4, C22×D13, C2×C13⋊D4
►On 104 points
Generators in S104
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 65)(27 92)(28 93)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 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 27 14 40)(2 39 15 52)(3 38 16 51)(4 37 17 50)(5 36 18 49)(6 35 19 48)(7 34 20 47)(8 33 21 46)(9 32 22 45)(10 31 23 44)(11 30 24 43)(12 29 25 42)(13 28 26 41)(53 79 66 92)(54 91 67 104)(55 90 68 103)(56 89 69 102)(57 88 70 101)(58 87 71 100)(59 86 72 99)(60 85 73 98)(61 84 74 97)(62 83 75 96)(63 82 76 95)(64 81 77 94)(65 80 78 93)
(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(27 40)(28 52)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(54 65)(55 64)(56 63)(57 62)(58 61)(59 60)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(79 92)(80 104)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)
G:=sub<Sym(104)| (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91), (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), (1,27,14,40)(2,39,15,52)(3,38,16,51)(4,37,17,50)(5,36,18,49)(6,35,19,48)(7,34,20,47)(8,33,21,46)(9,32,22,45)(10,31,23,44)(11,30,24,43)(12,29,25,42)(13,28,26,41)(53,79,66,92)(54,91,67,104)(55,90,68,103)(56,89,69,102)(57,88,70,101)(58,87,71,100)(59,86,72,99)(60,85,73,98)(61,84,74,97)(62,83,75,96)(63,82,76,95)(64,81,77,94)(65,80,78,93), (2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(27,40)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(79,92)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)>;
G:=Group( (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91), (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), (1,27,14,40)(2,39,15,52)(3,38,16,51)(4,37,17,50)(5,36,18,49)(6,35,19,48)(7,34,20,47)(8,33,21,46)(9,32,22,45)(10,31,23,44)(11,30,24,43)(12,29,25,42)(13,28,26,41)(53,79,66,92)(54,91,67,104)(55,90,68,103)(56,89,69,102)(57,88,70,101)(58,87,71,100)(59,86,72,99)(60,85,73,98)(61,84,74,97)(62,83,75,96)(63,82,76,95)(64,81,77,94)(65,80,78,93), (2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(27,40)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(79,92)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93) );
G=PermutationGroup([[(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,73),(9,74),(10,75),(11,76),(12,77),(13,78),(14,53),(15,54),(16,55),(17,56),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,64),(26,65),(27,92),(28,93),(29,94),(30,95),(31,96),(32,97),(33,98),(34,99),(35,100),(36,101),(37,102),(38,103),(39,104),(40,79),(41,80),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,88),(50,89),(51,90),(52,91)], [(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)], [(1,27,14,40),(2,39,15,52),(3,38,16,51),(4,37,17,50),(5,36,18,49),(6,35,19,48),(7,34,20,47),(8,33,21,46),(9,32,22,45),(10,31,23,44),(11,30,24,43),(12,29,25,42),(13,28,26,41),(53,79,66,92),(54,91,67,104),(55,90,68,103),(56,89,69,102),(57,88,70,101),(58,87,71,100),(59,86,72,99),(60,85,73,98),(61,84,74,97),(62,83,75,96),(63,82,76,95),(64,81,77,94),(65,80,78,93)], [(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(27,40),(28,52),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(54,65),(55,64),(56,63),(57,62),(58,61),(59,60),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73),(79,92),(80,104),(81,103),(82,102),(83,101),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93)]])
C2×C13⋊D4 is a maximal subgroup of
C22.2D52 D26.4D4 Dic13.4D4 Dic13⋊4D4 C22⋊D52 D26.12D4 D26⋊D4 C23.6D26 C22.D52 C23.23D26 C52⋊7D4 C23⋊D26 C52⋊2D4 Dic13⋊D4 C52⋊D4 C24⋊D13 C2×D4×D13 D4⋊6D26
C2×C13⋊D4 is a maximal quotient of
C52.48D4 C23.23D26 C52⋊7D4 D52⋊6C22 C23.18D26 C52.17D4 C23⋊D26 C52⋊2D4 Dic13⋊D4 C52⋊D4 Q8.D26 Dic13⋊Q8 D26⋊3Q8 C52.23D4 D4⋊D26 C52.C23 D4.9D26 C24⋊D13
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 13A | ··· | 13F | 26A | ··· | 26AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D13 | D26 | C13⋊D4 |
| kernel | C2×C13⋊D4 | C2×Dic13 | C13⋊D4 | C22×D13 | C22×C26 | C26 | C23 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 6 | 18 | 24 |
Matrix representation of C2×C13⋊D4 ►in GL3(𝔽53) generated by
| 52 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 39 | 52 |
| 0 | 31 | 40 |
| 1 | 0 | 0 |
| 0 | 13 | 29 |
| 0 | 38 | 40 |
| 52 | 0 | 0 |
| 0 | 6 | 27 |
| 0 | 36 | 47 |
G:=sub<GL(3,GF(53))| [52,0,0,0,1,0,0,0,1],[1,0,0,0,39,31,0,52,40],[1,0,0,0,13,38,0,29,40],[52,0,0,0,6,36,0,27,47] >;
C2×C13⋊D4 in GAP, Magma, Sage, TeX
C_2\times C_{13}\rtimes D_4 % in TeX
G:=Group("C2xC13:D4"); // GroupNames label
G:=SmallGroup(208,44);
// by ID
G=gap.SmallGroup(208,44);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,182,4804]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^13=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations