metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D52⋊5C2, C4.16D26, Dic26⋊5C2, C26.4C23, C22.2D26, C52.16C22, D26.1C22, Dic13.2C22, (C2×C52)⋊4C2, (C2×C4)⋊3D13, (C4×D13)⋊4C2, C13⋊1(C4○D4), C13⋊D4⋊3C2, C2.5(C22×D13), (C2×C26).11C22, SmallGroup(208,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D52⋊5C2
G = < a,b,c | a52=b2=c2=1, bab=a-1, ac=ca, cbc=a26b >
Smallest permutation representation of D52⋊5C2
►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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(61 63)(72 104)(73 103)(74 102)(75 101)(76 100)(77 99)(78 98)(79 97)(80 96)(81 95)(82 94)(83 93)(84 92)(85 91)(86 90)(87 89)
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 81)
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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(61,63)(72,104)(73,103)(74,102)(75,101)(76,100)(77,99)(78,98)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)(85,91)(86,90)(87,89), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81)>;
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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(61,63)(72,104)(73,103)(74,102)(75,101)(76,100)(77,99)(78,98)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)(85,91)(86,90)(87,89), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81) );
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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,24),(17,23),(18,22),(19,21),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(53,71),(54,70),(55,69),(56,68),(57,67),(58,66),(59,65),(60,64),(61,63),(72,104),(73,103),(74,102),(75,101),(76,100),(77,99),(78,98),(79,97),(80,96),(81,95),(82,94),(83,93),(84,92),(85,91),(86,90),(87,89)], [(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,91),(11,92),(12,93),(13,94),(14,95),(15,96),(16,97),(17,98),(18,99),(19,100),(20,101),(21,102),(22,103),(23,104),(24,53),(25,54),(26,55),(27,56),(28,57),(29,58),(30,59),(31,60),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67),(39,68),(40,69),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79),(51,80),(52,81)]])
D52⋊5C2 is a maximal subgroup of
D52⋊4C4 D52⋊7C4 D52.3C4 D104⋊7C2 D52.2C4 C8⋊D26 C8.D26 D52⋊6C22 Q8.D26 C52.C23 D4⋊6D26 Q8.10D26 C4○D4×D13 D4⋊8D26 D4.10D26
D52⋊5C2 is a maximal quotient of
C4×Dic26 C52.6Q8 C42⋊D13 C4×D52 C4.D52 C42⋊2D13 C23.D26 D26.12D4 D26⋊D4 C23.6D26 Dic13.Q8 D26.13D4 D26⋊Q8 C4⋊C4⋊D13 C52.48D4 C23.21D26 C4×C13⋊D4 C23.23D26 C52⋊7D4
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 26 | 26 | 1 | 1 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4○D4 | D13 | D26 | D26 | D52⋊5C2 |
| kernel | D52⋊5C2 | Dic26 | C4×D13 | D52 | C13⋊D4 | C2×C52 | C13 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 6 | 12 | 6 | 24 |
Matrix representation of D52⋊5C2 ►in GL2(𝔽53) generated by
| 33 | 40 |
| 27 | 44 |
| 9 | 52 |
| 27 | 44 |
| 28 | 18 |
| 36 | 25 |
G:=sub<GL(2,GF(53))| [33,27,40,44],[9,27,52,44],[28,36,18,25] >;
D52⋊5C2 in GAP, Magma, Sage, TeX
D_{52}\rtimes_5C_2 % in TeX
G:=Group("D52:5C2"); // GroupNames label
G:=SmallGroup(208,38);
// by ID
G=gap.SmallGroup(208,38);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,46,182,4804]);
// Polycyclic
G:=Group<a,b,c|a^52=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^26*b>;
// generators/relations
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