metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D52⋊7C4, C52.54D4, Dic26⋊7C4, C22.3D52, M4(2)⋊4D13, C13⋊4C4≀C2, C4.3(C4×D13), (C2×C26).1D4, C52.27(C2×C4), (C2×C4).38D26, (C4×Dic13)⋊1C2, D52⋊5C2.2C2, C4.29(C13⋊D4), (C13×M4(2))⋊8C2, (C2×C52).15C22, C26.21(C22⋊C4), C2.11(D26⋊C4), SmallGroup(416,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D52⋊7C4
G = < a,b,c | a52=b2=c4=1, bab=a-1, cac-1=a25, cbc-1=a11b >
2C2
52C2
26C4
26C22
26C4
26C4
2C26
4D13
2C8
13D4
13Q8
26C2×C4
26C2×C4
26D4
2D26
13C42
13C4○D4
2C104
13C4≀C2
Smallest permutation representation of D52⋊7C4
►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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 53)(26 104)(27 103)(28 102)(29 101)(30 100)(31 99)(32 98)(33 97)(34 96)(35 95)(36 94)(37 93)(38 92)(39 91)(40 90)(41 89)(42 88)(43 87)(44 86)(45 85)(46 84)(47 83)(48 82)(49 81)(50 80)(51 79)(52 78)
(1 40 27 14)(2 13 28 39)(3 38 29 12)(4 11 30 37)(5 36 31 10)(6 9 32 35)(7 34 33 8)(15 26 41 52)(16 51 42 25)(17 24 43 50)(18 49 44 23)(19 22 45 48)(20 47 46 21)(53 77)(54 102)(55 75)(56 100)(57 73)(58 98)(59 71)(60 96)(61 69)(62 94)(63 67)(64 92)(66 90)(68 88)(70 86)(72 84)(74 82)(76 80)(79 103)(81 101)(83 99)(85 97)(87 95)(89 93)
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,104)(27,103)(28,102)(29,101)(30,100)(31,99)(32,98)(33,97)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,80)(51,79)(52,78), (1,40,27,14)(2,13,28,39)(3,38,29,12)(4,11,30,37)(5,36,31,10)(6,9,32,35)(7,34,33,8)(15,26,41,52)(16,51,42,25)(17,24,43,50)(18,49,44,23)(19,22,45,48)(20,47,46,21)(53,77)(54,102)(55,75)(56,100)(57,73)(58,98)(59,71)(60,96)(61,69)(62,94)(63,67)(64,92)(66,90)(68,88)(70,86)(72,84)(74,82)(76,80)(79,103)(81,101)(83,99)(85,97)(87,95)(89,93)>;
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,104)(27,103)(28,102)(29,101)(30,100)(31,99)(32,98)(33,97)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,80)(51,79)(52,78), (1,40,27,14)(2,13,28,39)(3,38,29,12)(4,11,30,37)(5,36,31,10)(6,9,32,35)(7,34,33,8)(15,26,41,52)(16,51,42,25)(17,24,43,50)(18,49,44,23)(19,22,45,48)(20,47,46,21)(53,77)(54,102)(55,75)(56,100)(57,73)(58,98)(59,71)(60,96)(61,69)(62,94)(63,67)(64,92)(66,90)(68,88)(70,86)(72,84)(74,82)(76,80)(79,103)(81,101)(83,99)(85,97)(87,95)(89,93) );
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)], [(1,77),(2,76),(3,75),(4,74),(5,73),(6,72),(7,71),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,63),(16,62),(17,61),(18,60),(19,59),(20,58),(21,57),(22,56),(23,55),(24,54),(25,53),(26,104),(27,103),(28,102),(29,101),(30,100),(31,99),(32,98),(33,97),(34,96),(35,95),(36,94),(37,93),(38,92),(39,91),(40,90),(41,89),(42,88),(43,87),(44,86),(45,85),(46,84),(47,83),(48,82),(49,81),(50,80),(51,79),(52,78)], [(1,40,27,14),(2,13,28,39),(3,38,29,12),(4,11,30,37),(5,36,31,10),(6,9,32,35),(7,34,33,8),(15,26,41,52),(16,51,42,25),(17,24,43,50),(18,49,44,23),(19,22,45,48),(20,47,46,21),(53,77),(54,102),(55,75),(56,100),(57,73),(58,98),(59,71),(60,96),(61,69),(62,94),(63,67),(64,92),(66,90),(68,88),(70,86),(72,84),(74,82),(76,80),(79,103),(81,101),(83,99),(85,97),(87,95),(89,93)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26L | 52A | ··· | 52L | 52M | ··· | 52R | 104A | ··· | 104X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 52 | 1 | 1 | 2 | 26 | 26 | 26 | 26 | 52 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D13 | C4≀C2 | D26 | C4×D13 | C13⋊D4 | D52 | D52⋊7C4 |
| kernel | D52⋊7C4 | C4×Dic13 | C13×M4(2) | D52⋊5C2 | Dic26 | D52 | C52 | C2×C26 | M4(2) | C13 | C2×C4 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 6 | 4 | 6 | 12 | 12 | 12 | 12 |
Matrix representation of D52⋊7C4 ►in GL4(𝔽313) generated by
| 25 | 140 | 0 | 0 |
| 0 | 288 | 0 | 0 |
| 0 | 0 | 135 | 227 |
| 0 | 0 | 86 | 293 |
| 312 | 0 | 0 | 0 |
| 291 | 1 | 0 | 0 |
| 0 | 0 | 212 | 243 |
| 0 | 0 | 101 | 101 |
| 288 | 115 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 86 | 293 |
| 0 | 0 | 135 | 227 |
G:=sub<GL(4,GF(313))| [25,0,0,0,140,288,0,0,0,0,135,86,0,0,227,293],[312,291,0,0,0,1,0,0,0,0,212,101,0,0,243,101],[288,0,0,0,115,1,0,0,0,0,86,135,0,0,293,227] >;
D52⋊7C4 in GAP, Magma, Sage, TeX
D_{52}\rtimes_7C_4 % in TeX
G:=Group("D52:7C4"); // GroupNames label
G:=SmallGroup(416,32);
// by ID
G=gap.SmallGroup(416,32);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,121,31,86,579,297,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^25,c*b*c^-1=a^11*b>;
// generators/relations
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