metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊2D4, D4⋊2D28, D14⋊4D8, C4⋊C4⋊1D14, (C7×D4)⋊1D4, (C2×C8)⋊2D14, C2.8(D7×D8), D14⋊C8⋊4C2, (C2×D56)⋊4C2, C4.84(D4×D7), C4.1(C2×D28), D4⋊C4⋊4D7, C4⋊D28⋊1C2, C7⋊2(C22⋊D8), (C2×C56)⋊2C22, C14.22(C2×D8), C14.D8⋊6C2, C28.107(C2×D4), C14.19C22≀C2, (C2×D4).134D14, (C2×D28)⋊12C22, (C2×Dic7).18D4, (C22×D7).70D4, C22.171(D4×D7), C2.10(D56⋊C2), C14.55(C8⋊C22), (C2×C28).213C23, (D4×C14).34C22, C2.22(C22⋊D28), (C2×D4×D7)⋊1C2, (C2×D4⋊D7)⋊2C2, (C2×C7⋊C8)⋊2C22, (C7×C4⋊C4)⋊3C22, (C7×D4⋊C4)⋊4C2, (C2×C4×D7).7C22, (C2×C14).226(C2×D4), (C2×C4).320(C22×D7), SmallGroup(448,307)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D28
G = < a,b,c,d | a4=b2=c28=d2=1, bab=cac-1=dad=a-1, cbc-1=dbd=a-1b, dcd=c-1 >
Subgroups: 1524 in 198 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, D4⋊C4, C4⋊D4, C2×D8, C22×D4, C7⋊C8, C56, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C22×D7, C22×D7, C22×C14, C22⋊D8, D56, C2×C7⋊C8, D14⋊C4, D4⋊D7, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×D28, C2×D28, D4×D7, C2×C7⋊D4, D4×C14, C23×D7, C14.D8, D14⋊C8, C7×D4⋊C4, C4⋊D28, C2×D56, C2×D4⋊D7, C2×D4×D7, D4⋊D28
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C22≀C2, C2×D8, C8⋊C22, D28, C22×D7, C22⋊D8, C2×D28, D4×D7, C22⋊D28, D7×D8, D56⋊C2, D4⋊D28
►On 112 points
Generators in S112
(1 72 56 109)(2 110 29 73)(3 74 30 111)(4 112 31 75)(5 76 32 85)(6 86 33 77)(7 78 34 87)(8 88 35 79)(9 80 36 89)(10 90 37 81)(11 82 38 91)(12 92 39 83)(13 84 40 93)(14 94 41 57)(15 58 42 95)(16 96 43 59)(17 60 44 97)(18 98 45 61)(19 62 46 99)(20 100 47 63)(21 64 48 101)(22 102 49 65)(23 66 50 103)(24 104 51 67)(25 68 52 105)(26 106 53 69)(27 70 54 107)(28 108 55 71)
(1 42)(2 96)(3 44)(4 98)(5 46)(6 100)(7 48)(8 102)(9 50)(10 104)(11 52)(12 106)(13 54)(14 108)(15 56)(16 110)(17 30)(18 112)(19 32)(20 86)(21 34)(22 88)(23 36)(24 90)(25 38)(26 92)(27 40)(28 94)(29 59)(31 61)(33 63)(35 65)(37 67)(39 69)(41 71)(43 73)(45 75)(47 77)(49 79)(51 81)(53 83)(55 57)(58 72)(60 74)(62 76)(64 78)(66 80)(68 82)(70 84)(85 99)(87 101)(89 103)(91 105)(93 107)(95 109)(97 111)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(28 56)(57 95)(58 94)(59 93)(60 92)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 112)(69 111)(70 110)(71 109)(72 108)(73 107)(74 106)(75 105)(76 104)(77 103)(78 102)(79 101)(80 100)(81 99)(82 98)(83 97)(84 96)
G:=sub<Sym(112)| (1,72,56,109)(2,110,29,73)(3,74,30,111)(4,112,31,75)(5,76,32,85)(6,86,33,77)(7,78,34,87)(8,88,35,79)(9,80,36,89)(10,90,37,81)(11,82,38,91)(12,92,39,83)(13,84,40,93)(14,94,41,57)(15,58,42,95)(16,96,43,59)(17,60,44,97)(18,98,45,61)(19,62,46,99)(20,100,47,63)(21,64,48,101)(22,102,49,65)(23,66,50,103)(24,104,51,67)(25,68,52,105)(26,106,53,69)(27,70,54,107)(28,108,55,71), (1,42)(2,96)(3,44)(4,98)(5,46)(6,100)(7,48)(8,102)(9,50)(10,104)(11,52)(12,106)(13,54)(14,108)(15,56)(16,110)(17,30)(18,112)(19,32)(20,86)(21,34)(22,88)(23,36)(24,90)(25,38)(26,92)(27,40)(28,94)(29,59)(31,61)(33,63)(35,65)(37,67)(39,69)(41,71)(43,73)(45,75)(47,77)(49,79)(51,81)(53,83)(55,57)(58,72)(60,74)(62,76)(64,78)(66,80)(68,82)(70,84)(85,99)(87,101)(89,103)(91,105)(93,107)(95,109)(97,111), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(28,56)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,112)(69,111)(70,110)(71,109)(72,108)(73,107)(74,106)(75,105)(76,104)(77,103)(78,102)(79,101)(80,100)(81,99)(82,98)(83,97)(84,96)>;
G:=Group( (1,72,56,109)(2,110,29,73)(3,74,30,111)(4,112,31,75)(5,76,32,85)(6,86,33,77)(7,78,34,87)(8,88,35,79)(9,80,36,89)(10,90,37,81)(11,82,38,91)(12,92,39,83)(13,84,40,93)(14,94,41,57)(15,58,42,95)(16,96,43,59)(17,60,44,97)(18,98,45,61)(19,62,46,99)(20,100,47,63)(21,64,48,101)(22,102,49,65)(23,66,50,103)(24,104,51,67)(25,68,52,105)(26,106,53,69)(27,70,54,107)(28,108,55,71), (1,42)(2,96)(3,44)(4,98)(5,46)(6,100)(7,48)(8,102)(9,50)(10,104)(11,52)(12,106)(13,54)(14,108)(15,56)(16,110)(17,30)(18,112)(19,32)(20,86)(21,34)(22,88)(23,36)(24,90)(25,38)(26,92)(27,40)(28,94)(29,59)(31,61)(33,63)(35,65)(37,67)(39,69)(41,71)(43,73)(45,75)(47,77)(49,79)(51,81)(53,83)(55,57)(58,72)(60,74)(62,76)(64,78)(66,80)(68,82)(70,84)(85,99)(87,101)(89,103)(91,105)(93,107)(95,109)(97,111), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(28,56)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,112)(69,111)(70,110)(71,109)(72,108)(73,107)(74,106)(75,105)(76,104)(77,103)(78,102)(79,101)(80,100)(81,99)(82,98)(83,97)(84,96) );
G=PermutationGroup([[(1,72,56,109),(2,110,29,73),(3,74,30,111),(4,112,31,75),(5,76,32,85),(6,86,33,77),(7,78,34,87),(8,88,35,79),(9,80,36,89),(10,90,37,81),(11,82,38,91),(12,92,39,83),(13,84,40,93),(14,94,41,57),(15,58,42,95),(16,96,43,59),(17,60,44,97),(18,98,45,61),(19,62,46,99),(20,100,47,63),(21,64,48,101),(22,102,49,65),(23,66,50,103),(24,104,51,67),(25,68,52,105),(26,106,53,69),(27,70,54,107),(28,108,55,71)], [(1,42),(2,96),(3,44),(4,98),(5,46),(6,100),(7,48),(8,102),(9,50),(10,104),(11,52),(12,106),(13,54),(14,108),(15,56),(16,110),(17,30),(18,112),(19,32),(20,86),(21,34),(22,88),(23,36),(24,90),(25,38),(26,92),(27,40),(28,94),(29,59),(31,61),(33,63),(35,65),(37,67),(39,69),(41,71),(43,73),(45,75),(47,77),(49,79),(51,81),(53,83),(55,57),(58,72),(60,74),(62,76),(64,78),(66,80),(68,82),(70,84),(85,99),(87,101),(89,103),(91,105),(93,107),(95,109),(97,111)], [(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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,48),(9,47),(10,46),(11,45),(12,44),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(28,56),(57,95),(58,94),(59,93),(60,92),(61,91),(62,90),(63,89),(64,88),(65,87),(66,86),(67,85),(68,112),(69,111),(70,110),(71,109),(72,108),(73,107),(74,106),(75,105),(76,104),(77,103),(78,102),(79,101),(80,100),(81,99),(82,98),(83,97),(84,96)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 28 | 28 | 56 | 2 | 2 | 8 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D7 | D8 | D14 | D14 | D14 | D28 | C8⋊C22 | D4×D7 | D4×D7 | D7×D8 | D56⋊C2 |
| kernel | D4⋊D28 | C14.D8 | D14⋊C8 | C7×D4⋊C4 | C4⋊D28 | C2×D56 | C2×D4⋊D7 | C2×D4×D7 | D28 | C2×Dic7 | C7×D4 | C22×D7 | D4⋊C4 | D14 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 4 | 3 | 3 | 3 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of D4⋊D28 ►in GL4(𝔽113) generated by
| 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 82 | 31 | 0 | 0 |
| 31 | 31 | 0 | 0 |
| 0 | 0 | 4 | 81 |
| 0 | 0 | 32 | 55 |
| 31 | 82 | 0 | 0 |
| 82 | 82 | 0 | 0 |
| 0 | 0 | 109 | 32 |
| 0 | 0 | 109 | 4 |
G:=sub<GL(4,GF(113))| [0,1,0,0,112,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,112,0,0,0,0,112],[82,31,0,0,31,31,0,0,0,0,4,32,0,0,81,55],[31,82,0,0,82,82,0,0,0,0,109,109,0,0,32,4] >;
D4⋊D28 in GAP, Magma, Sage, TeX
D_4\rtimes D_{28} % in TeX
G:=Group("D4:D28"); // GroupNames label
G:=SmallGroup(448,307);
// by ID
G=gap.SmallGroup(448,307);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,58,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^28=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations