direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D52, C4⋊2D26, C26⋊1D4, C52⋊2C22, D26⋊1C22, C26.3C23, C22.10D26, C13⋊1(C2×D4), (C2×C52)⋊3C2, (C2×C4)⋊2D13, (C22×D13)⋊1C2, C2.4(C22×D13), (C2×C26).10C22, SmallGroup(208,37)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D52
G = < a,b,c | a2=b52=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 394 in 54 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, C23, C13, C2×D4, D13, C26, C26, C52, D26, D26, C2×C26, D52, C2×C52, C22×D13, C2×D52
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, D26, D52, C22×D13, C2×D52
►On 104 points
Generators in S104
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 104)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 85)(51 86)(52 87)
(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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(61 62)(71 104)(72 103)(73 102)(74 101)(75 100)(76 99)(77 98)(78 97)(79 96)(80 95)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)
G:=sub<Sym(104)| (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87), (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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)>;
G:=Group( (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87), (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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88) );
G=PermutationGroup([[(1,88),(2,89),(3,90),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,97),(11,98),(12,99),(13,100),(14,101),(15,102),(16,103),(17,104),(18,53),(19,54),(20,55),(21,56),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71),(37,72),(38,73),(39,74),(40,75),(41,76),(42,77),(43,78),(44,79),(45,80),(46,81),(47,82),(48,83),(49,84),(50,85),(51,86),(52,87)], [(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)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,36),(18,35),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(53,70),(54,69),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63),(61,62),(71,104),(72,103),(73,102),(74,101),(75,100),(76,99),(77,98),(78,97),(79,96),(80,95),(81,94),(82,93),(83,92),(84,91),(85,90),(86,89),(87,88)]])
C2×D52 is a maximal subgroup of
D52⋊6C4 D52⋊5C4 C52.46D4 D26.D4 C4⋊D52 C4.D52 C22⋊D52 D26⋊D4 D52⋊8C4 D26.13D4 C4⋊2D52 C8⋊D26 C52⋊7D4 C52⋊D4 C52.23D4 D4⋊D26 C2×D4×D13 D4⋊8D26
C2×D52 is a maximal quotient of
C52⋊2Q8 C4⋊D52 C4.D52 C22⋊D52 C22.D52 C4⋊2D52 D26⋊2Q8 D104⋊7C2 C8⋊D26 C8.D26 C52⋊7D4
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 26 | 26 | 26 | 26 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D4 | D13 | D26 | D26 | D52 |
| kernel | C2×D52 | D52 | C2×C52 | C22×D13 | C26 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 6 | 12 | 6 | 24 |
Matrix representation of C2×D52 ►in GL4(𝔽53) generated by
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 10 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 33 | 38 |
| 0 | 0 | 0 | 45 |
| 15 | 1 | 0 | 0 |
| 41 | 38 | 0 | 0 |
| 0 | 0 | 27 | 18 |
| 0 | 0 | 42 | 26 |
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,10,28,0,0,0,0,33,0,0,0,38,45],[15,41,0,0,1,38,0,0,0,0,27,42,0,0,18,26] >;
C2×D52 in GAP, Magma, Sage, TeX
C_2\times D_{52} % in TeX
G:=Group("C2xD52"); // GroupNames label
G:=SmallGroup(208,37);
// by ID
G=gap.SmallGroup(208,37);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,182,42,4804]);
// Polycyclic
G:=Group<a,b,c|a^2=b^52=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations