metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D26⋊4D4, C22⋊2D52, C23.15D26, (C2×C4)⋊1D26, (C2×C26)⋊1D4, C13⋊1C22≀C2, (C2×D52)⋊2C2, C2.7(C2×D52), C26.5(C2×D4), C2.7(D4×D13), (C2×C52)⋊1C22, C22⋊C4⋊2D13, D26⋊C4⋊4C2, (C23×D13)⋊1C2, (C2×C26).23C23, (C2×Dic13)⋊1C22, (C22×D13)⋊1C22, (C22×C26).12C22, C22.41(C22×D13), (C2×C13⋊D4)⋊1C2, (C13×C22⋊C4)⋊3C2, SmallGroup(416,103)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊D52
G = < a,b,c,d | a2=b2=c52=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1208 in 130 conjugacy classes, 37 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C13, C22⋊C4, C22⋊C4, C2×D4, C24, D13, C26, C26, C26, C22≀C2, Dic13, C52, D26, D26, C2×C26, C2×C26, C2×C26, D52, C2×Dic13, C13⋊D4, C2×C52, C22×D13, C22×D13, C22×D13, C22×C26, D26⋊C4, C13×C22⋊C4, C2×D52, C2×C13⋊D4, C23×D13, C22⋊D52
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C22≀C2, D26, D52, C22×D13, C2×D52, D4×D13, C22⋊D52
►On 104 points
Generators in S104
(1 27)(2 71)(3 29)(4 73)(5 31)(6 75)(7 33)(8 77)(9 35)(10 79)(11 37)(12 81)(13 39)(14 83)(15 41)(16 85)(17 43)(18 87)(19 45)(20 89)(21 47)(22 91)(23 49)(24 93)(25 51)(26 95)(28 97)(30 99)(32 101)(34 103)(36 53)(38 55)(40 57)(42 59)(44 61)(46 63)(48 65)(50 67)(52 69)(54 80)(56 82)(58 84)(60 86)(62 88)(64 90)(66 92)(68 94)(70 96)(72 98)(74 100)(76 102)(78 104)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 92)(50 93)(51 94)(52 95)
(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 95)(2 94)(3 93)(4 92)(5 91)(6 90)(7 89)(8 88)(9 87)(10 86)(11 85)(12 84)(13 83)(14 82)(15 81)(16 80)(17 79)(18 78)(19 77)(20 76)(21 75)(22 74)(23 73)(24 72)(25 71)(26 70)(27 69)(28 68)(29 67)(30 66)(31 65)(32 64)(33 63)(34 62)(35 61)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 104)(45 103)(46 102)(47 101)(48 100)(49 99)(50 98)(51 97)(52 96)
G:=sub<Sym(104)| (1,27)(2,71)(3,29)(4,73)(5,31)(6,75)(7,33)(8,77)(9,35)(10,79)(11,37)(12,81)(13,39)(14,83)(15,41)(16,85)(17,43)(18,87)(19,45)(20,89)(21,47)(22,91)(23,49)(24,93)(25,51)(26,95)(28,97)(30,99)(32,101)(34,103)(36,53)(38,55)(40,57)(42,59)(44,61)(46,63)(48,65)(50,67)(52,69)(54,80)(56,82)(58,84)(60,86)(62,88)(64,90)(66,92)(68,94)(70,96)(72,98)(74,100)(76,102)(78,104), (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95), (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), (1,95)(2,94)(3,93)(4,92)(5,91)(6,90)(7,89)(8,88)(9,87)(10,86)(11,85)(12,84)(13,83)(14,82)(15,81)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,74)(23,73)(24,72)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,104)(45,103)(46,102)(47,101)(48,100)(49,99)(50,98)(51,97)(52,96)>;
G:=Group( (1,27)(2,71)(3,29)(4,73)(5,31)(6,75)(7,33)(8,77)(9,35)(10,79)(11,37)(12,81)(13,39)(14,83)(15,41)(16,85)(17,43)(18,87)(19,45)(20,89)(21,47)(22,91)(23,49)(24,93)(25,51)(26,95)(28,97)(30,99)(32,101)(34,103)(36,53)(38,55)(40,57)(42,59)(44,61)(46,63)(48,65)(50,67)(52,69)(54,80)(56,82)(58,84)(60,86)(62,88)(64,90)(66,92)(68,94)(70,96)(72,98)(74,100)(76,102)(78,104), (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95), (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), (1,95)(2,94)(3,93)(4,92)(5,91)(6,90)(7,89)(8,88)(9,87)(10,86)(11,85)(12,84)(13,83)(14,82)(15,81)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,74)(23,73)(24,72)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,104)(45,103)(46,102)(47,101)(48,100)(49,99)(50,98)(51,97)(52,96) );
G=PermutationGroup([[(1,27),(2,71),(3,29),(4,73),(5,31),(6,75),(7,33),(8,77),(9,35),(10,79),(11,37),(12,81),(13,39),(14,83),(15,41),(16,85),(17,43),(18,87),(19,45),(20,89),(21,47),(22,91),(23,49),(24,93),(25,51),(26,95),(28,97),(30,99),(32,101),(34,103),(36,53),(38,55),(40,57),(42,59),(44,61),(46,63),(48,65),(50,67),(52,69),(54,80),(56,82),(58,84),(60,86),(62,88),(64,90),(66,92),(68,94),(70,96),(72,98),(74,100),(76,102),(78,104)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,101),(7,102),(8,103),(9,104),(10,53),(11,54),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,61),(19,62),(20,63),(21,64),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(28,71),(29,72),(30,73),(31,74),(32,75),(33,76),(34,77),(35,78),(36,79),(37,80),(38,81),(39,82),(40,83),(41,84),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,92),(50,93),(51,94),(52,95)], [(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)], [(1,95),(2,94),(3,93),(4,92),(5,91),(6,90),(7,89),(8,88),(9,87),(10,86),(11,85),(12,84),(13,83),(14,82),(15,81),(16,80),(17,79),(18,78),(19,77),(20,76),(21,75),(22,74),(23,73),(24,72),(25,71),(26,70),(27,69),(28,68),(29,67),(30,66),(31,65),(32,64),(33,63),(34,62),(35,61),(36,60),(37,59),(38,58),(39,57),(40,56),(41,55),(42,54),(43,53),(44,104),(45,103),(46,102),(47,101),(48,100),(49,99),(50,98),(51,97),(52,96)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 13A | ··· | 13F | 26A | ··· | 26R | 26S | ··· | 26AD | 52A | ··· | 52X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 26 | 26 | 26 | 26 | 52 | 4 | 4 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D13 | D26 | D26 | D52 | D4×D13 |
| kernel | C22⋊D52 | D26⋊C4 | C13×C22⋊C4 | C2×D52 | C2×C13⋊D4 | C23×D13 | D26 | C2×C26 | C22⋊C4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 2 | 6 | 12 | 6 | 24 | 12 |
Matrix representation of C22⋊D52 ►in GL4(𝔽53) generated by
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 29 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 28 | 18 | 0 | 0 |
| 45 | 46 | 0 | 0 |
| 0 | 0 | 21 | 38 |
| 0 | 0 | 33 | 32 |
| 3 | 23 | 0 | 0 |
| 25 | 50 | 0 | 0 |
| 0 | 0 | 32 | 15 |
| 0 | 0 | 6 | 21 |
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,52,29,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[28,45,0,0,18,46,0,0,0,0,21,33,0,0,38,32],[3,25,0,0,23,50,0,0,0,0,32,6,0,0,15,21] >;
C22⋊D52 in GAP, Magma, Sage, TeX
C_2^2\rtimes D_{52} % in TeX
G:=Group("C2^2:D52"); // GroupNames label
G:=SmallGroup(416,103);
// by ID
G=gap.SmallGroup(416,103);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,218,188,50,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^52=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations