metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D52⋊4C2, Q8⋊2D13, C4.7D26, C26.8C23, C52.7C22, D26.3C22, Dic13.9C22, (C4×D13)⋊3C2, C13⋊3(C4○D4), (Q8×C13)⋊3C2, C2.9(C22×D13), SmallGroup(208,42)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D52⋊C2
G = < a,b,c | a52=b2=c2=1, bab=a-1, cac=a25, cbc=a50b >
Smallest permutation representation of D52⋊C2
►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 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 58)(54 57)(55 56)(59 104)(60 103)(61 102)(62 101)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 84)(80 83)(81 82)
(1 69)(2 94)(3 67)(4 92)(5 65)(6 90)(7 63)(8 88)(9 61)(10 86)(11 59)(12 84)(13 57)(14 82)(15 55)(16 80)(17 53)(18 78)(19 103)(20 76)(21 101)(22 74)(23 99)(24 72)(25 97)(26 70)(27 95)(28 68)(29 93)(30 66)(31 91)(32 64)(33 89)(34 62)(35 87)(36 60)(37 85)(38 58)(39 83)(40 56)(41 81)(42 54)(43 79)(44 104)(45 77)(46 102)(47 75)(48 100)(49 73)(50 98)(51 71)(52 96)
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), (1,52)(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)(53,58)(54,57)(55,56)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82), (1,69)(2,94)(3,67)(4,92)(5,65)(6,90)(7,63)(8,88)(9,61)(10,86)(11,59)(12,84)(13,57)(14,82)(15,55)(16,80)(17,53)(18,78)(19,103)(20,76)(21,101)(22,74)(23,99)(24,72)(25,97)(26,70)(27,95)(28,68)(29,93)(30,66)(31,91)(32,64)(33,89)(34,62)(35,87)(36,60)(37,85)(38,58)(39,83)(40,56)(41,81)(42,54)(43,79)(44,104)(45,77)(46,102)(47,75)(48,100)(49,73)(50,98)(51,71)(52,96)>;
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), (1,52)(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)(53,58)(54,57)(55,56)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82), (1,69)(2,94)(3,67)(4,92)(5,65)(6,90)(7,63)(8,88)(9,61)(10,86)(11,59)(12,84)(13,57)(14,82)(15,55)(16,80)(17,53)(18,78)(19,103)(20,76)(21,101)(22,74)(23,99)(24,72)(25,97)(26,70)(27,95)(28,68)(29,93)(30,66)(31,91)(32,64)(33,89)(34,62)(35,87)(36,60)(37,85)(38,58)(39,83)(40,56)(41,81)(42,54)(43,79)(44,104)(45,77)(46,102)(47,75)(48,100)(49,73)(50,98)(51,71)(52,96) );
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)], [(1,52),(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),(53,58),(54,57),(55,56),(59,104),(60,103),(61,102),(62,101),(63,100),(64,99),(65,98),(66,97),(67,96),(68,95),(69,94),(70,93),(71,92),(72,91),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85),(79,84),(80,83),(81,82)], [(1,69),(2,94),(3,67),(4,92),(5,65),(6,90),(7,63),(8,88),(9,61),(10,86),(11,59),(12,84),(13,57),(14,82),(15,55),(16,80),(17,53),(18,78),(19,103),(20,76),(21,101),(22,74),(23,99),(24,72),(25,97),(26,70),(27,95),(28,68),(29,93),(30,66),(31,91),(32,64),(33,89),(34,62),(35,87),(36,60),(37,85),(38,58),(39,83),(40,56),(41,81),(42,54),(43,79),(44,104),(45,77),(46,102),(47,75),(48,100),(49,73),(50,98),(51,71),(52,96)]])
D52⋊C2 is a maximal subgroup of
D52⋊C4 Q8⋊D26 D26.6D4 Q16⋊D13 D104⋊C2 D52.C4 Q8.10D26 C4○D4×D13 D4⋊8D26
D52⋊C2 is a maximal quotient of
C4.Dic26 C4⋊C4⋊7D13 D52⋊8C4 D26.13D4 C4⋊2D52 C4⋊C4⋊D13 Q8×Dic13 D26⋊3Q8 C52.23D4
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 13A | ··· | 13F | 26A | ··· | 26F | 52A | ··· | 52R |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 26 | 26 | 26 | 2 | 2 | 2 | 13 | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C4○D4 | D13 | D26 | D52⋊C2 |
| kernel | D52⋊C2 | C4×D13 | D52 | Q8×C13 | C13 | Q8 | C4 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 18 | 6 |
Matrix representation of D52⋊C2 ►in GL4(𝔽53) generated by
| 40 | 9 | 0 | 0 |
| 44 | 51 | 0 | 0 |
| 0 | 0 | 52 | 2 |
| 0 | 0 | 52 | 1 |
| 40 | 9 | 0 | 0 |
| 52 | 13 | 0 | 0 |
| 0 | 0 | 52 | 2 |
| 0 | 0 | 0 | 1 |
| 52 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 |
| 0 | 0 | 23 | 7 |
| 0 | 0 | 23 | 30 |
G:=sub<GL(4,GF(53))| [40,44,0,0,9,51,0,0,0,0,52,52,0,0,2,1],[40,52,0,0,9,13,0,0,0,0,52,0,0,0,2,1],[52,40,0,0,0,1,0,0,0,0,23,23,0,0,7,30] >;
D52⋊C2 in GAP, Magma, Sage, TeX
D_{52}\rtimes C_2 % in TeX
G:=Group("D52:C2"); // GroupNames label
G:=SmallGroup(208,42);
// by ID
G=gap.SmallGroup(208,42);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,46,182,97,42,4804]);
// Polycyclic
G:=Group<a,b,c|a^52=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^25,c*b*c=a^50*b>;
// generators/relations
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