metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊19D4, C14.1142+ 1+4, C7⋊3D42, C4⋊3(D4×D7), C28⋊6(C2×D4), C7⋊D4⋊1D4, C4⋊C4⋊23D14, D14⋊7(C2×D4), C22⋊2(D4×D7), (C2×D4)⋊23D14, C4⋊D4⋊12D7, Dic7⋊4(C2×D4), C4⋊D28⋊22C2, C28⋊D4⋊17C2, C22⋊C4⋊11D14, (C22×C4)⋊17D14, D28⋊C4⋊20C2, C23⋊D14⋊10C2, D14⋊D4⋊20C2, C22⋊D28⋊13C2, D14⋊C4⋊53C22, (C2×D28)⋊46C22, (C22×D28)⋊15C2, (D4×C14)⋊13C22, (C2×C28).40C23, C14.68(C22×D4), (C2×C14).153C24, Dic7⋊C4⋊52C22, (C22×C28)⋊21C22, (C4×Dic7)⋊22C22, (C23×D7)⋊10C22, C2.28(D4⋊8D14), C23.D7⋊51C22, (C22×D7).64C23, C22.174(C23×D7), C23.181(C22×D7), (C22×C14).188C23, (C2×Dic7).227C23, (C2×D4×D7)⋊11C2, C2.41(C2×D4×D7), (C2×C14)⋊3(C2×D4), (C4×C7⋊D4)⋊16C2, (C2×C4×D7)⋊14C22, (C7×C4⋊D4)⋊15C2, (C7×C4⋊C4)⋊11C22, (C2×C7⋊D4)⋊15C22, (C7×C22⋊C4)⋊13C22, (C2×C4).176(C22×D7), SmallGroup(448,1062)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28⋊19D4
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, cac-1=dad=a15, cbc-1=dbd=a14b, dcd=c-1 >
Subgroups: 2668 in 428 conjugacy classes, 115 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C4⋊1D4, C22×D4, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C22×C14, D42, C4×Dic7, Dic7⋊C4, D14⋊C4, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, C2×D28, D4×D7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C23×D7, C22⋊D28, D14⋊D4, D28⋊C4, C4⋊D28, C4×C7⋊D4, C23⋊D14, C28⋊D4, C7×C4⋊D4, C22×D28, C2×D4×D7, C2×D4×D7, D28⋊19D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, C22×D7, D42, D4×D7, C23×D7, C2×D4×D7, D4⋊8D14, D28⋊19D4
►On 112 points
(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 86)(2 85)(3 112)(4 111)(5 110)(6 109)(7 108)(8 107)(9 106)(10 105)(11 104)(12 103)(13 102)(14 101)(15 100)(16 99)(17 98)(18 97)(19 96)(20 95)(21 94)(22 93)(23 92)(24 91)(25 90)(26 89)(27 88)(28 87)(29 62)(30 61)(31 60)(32 59)(33 58)(34 57)(35 84)(36 83)(37 82)(38 81)(39 80)(40 79)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 72)(48 71)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)
(1 81 94 32)(2 68 95 47)(3 83 96 34)(4 70 97 49)(5 57 98 36)(6 72 99 51)(7 59 100 38)(8 74 101 53)(9 61 102 40)(10 76 103 55)(11 63 104 42)(12 78 105 29)(13 65 106 44)(14 80 107 31)(15 67 108 46)(16 82 109 33)(17 69 110 48)(18 84 111 35)(19 71 112 50)(20 58 85 37)(21 73 86 52)(22 60 87 39)(23 75 88 54)(24 62 89 41)(25 77 90 56)(26 64 91 43)(27 79 92 30)(28 66 93 45)
(1 67)(2 82)(3 69)(4 84)(5 71)(6 58)(7 73)(8 60)(9 75)(10 62)(11 77)(12 64)(13 79)(14 66)(15 81)(16 68)(17 83)(18 70)(19 57)(20 72)(21 59)(22 74)(23 61)(24 76)(25 63)(26 78)(27 65)(28 80)(29 91)(30 106)(31 93)(32 108)(33 95)(34 110)(35 97)(36 112)(37 99)(38 86)(39 101)(40 88)(41 103)(42 90)(43 105)(44 92)(45 107)(46 94)(47 109)(48 96)(49 111)(50 98)(51 85)(52 100)(53 87)(54 102)(55 89)(56 104)
G:=sub<Sym(112)| (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), (1,86)(2,85)(3,112)(4,111)(5,110)(6,109)(7,108)(8,107)(9,106)(10,105)(11,104)(12,103)(13,102)(14,101)(15,100)(16,99)(17,98)(18,97)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63), (1,81,94,32)(2,68,95,47)(3,83,96,34)(4,70,97,49)(5,57,98,36)(6,72,99,51)(7,59,100,38)(8,74,101,53)(9,61,102,40)(10,76,103,55)(11,63,104,42)(12,78,105,29)(13,65,106,44)(14,80,107,31)(15,67,108,46)(16,82,109,33)(17,69,110,48)(18,84,111,35)(19,71,112,50)(20,58,85,37)(21,73,86,52)(22,60,87,39)(23,75,88,54)(24,62,89,41)(25,77,90,56)(26,64,91,43)(27,79,92,30)(28,66,93,45), (1,67)(2,82)(3,69)(4,84)(5,71)(6,58)(7,73)(8,60)(9,75)(10,62)(11,77)(12,64)(13,79)(14,66)(15,81)(16,68)(17,83)(18,70)(19,57)(20,72)(21,59)(22,74)(23,61)(24,76)(25,63)(26,78)(27,65)(28,80)(29,91)(30,106)(31,93)(32,108)(33,95)(34,110)(35,97)(36,112)(37,99)(38,86)(39,101)(40,88)(41,103)(42,90)(43,105)(44,92)(45,107)(46,94)(47,109)(48,96)(49,111)(50,98)(51,85)(52,100)(53,87)(54,102)(55,89)(56,104)>;
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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,86)(2,85)(3,112)(4,111)(5,110)(6,109)(7,108)(8,107)(9,106)(10,105)(11,104)(12,103)(13,102)(14,101)(15,100)(16,99)(17,98)(18,97)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63), (1,81,94,32)(2,68,95,47)(3,83,96,34)(4,70,97,49)(5,57,98,36)(6,72,99,51)(7,59,100,38)(8,74,101,53)(9,61,102,40)(10,76,103,55)(11,63,104,42)(12,78,105,29)(13,65,106,44)(14,80,107,31)(15,67,108,46)(16,82,109,33)(17,69,110,48)(18,84,111,35)(19,71,112,50)(20,58,85,37)(21,73,86,52)(22,60,87,39)(23,75,88,54)(24,62,89,41)(25,77,90,56)(26,64,91,43)(27,79,92,30)(28,66,93,45), (1,67)(2,82)(3,69)(4,84)(5,71)(6,58)(7,73)(8,60)(9,75)(10,62)(11,77)(12,64)(13,79)(14,66)(15,81)(16,68)(17,83)(18,70)(19,57)(20,72)(21,59)(22,74)(23,61)(24,76)(25,63)(26,78)(27,65)(28,80)(29,91)(30,106)(31,93)(32,108)(33,95)(34,110)(35,97)(36,112)(37,99)(38,86)(39,101)(40,88)(41,103)(42,90)(43,105)(44,92)(45,107)(46,94)(47,109)(48,96)(49,111)(50,98)(51,85)(52,100)(53,87)(54,102)(55,89)(56,104) );
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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,86),(2,85),(3,112),(4,111),(5,110),(6,109),(7,108),(8,107),(9,106),(10,105),(11,104),(12,103),(13,102),(14,101),(15,100),(16,99),(17,98),(18,97),(19,96),(20,95),(21,94),(22,93),(23,92),(24,91),(25,90),(26,89),(27,88),(28,87),(29,62),(30,61),(31,60),(32,59),(33,58),(34,57),(35,84),(36,83),(37,82),(38,81),(39,80),(40,79),(41,78),(42,77),(43,76),(44,75),(45,74),(46,73),(47,72),(48,71),(49,70),(50,69),(51,68),(52,67),(53,66),(54,65),(55,64),(56,63)], [(1,81,94,32),(2,68,95,47),(3,83,96,34),(4,70,97,49),(5,57,98,36),(6,72,99,51),(7,59,100,38),(8,74,101,53),(9,61,102,40),(10,76,103,55),(11,63,104,42),(12,78,105,29),(13,65,106,44),(14,80,107,31),(15,67,108,46),(16,82,109,33),(17,69,110,48),(18,84,111,35),(19,71,112,50),(20,58,85,37),(21,73,86,52),(22,60,87,39),(23,75,88,54),(24,62,89,41),(25,77,90,56),(26,64,91,43),(27,79,92,30),(28,66,93,45)], [(1,67),(2,82),(3,69),(4,84),(5,71),(6,58),(7,73),(8,60),(9,75),(10,62),(11,77),(12,64),(13,79),(14,66),(15,81),(16,68),(17,83),(18,70),(19,57),(20,72),(21,59),(22,74),(23,61),(24,76),(25,63),(26,78),(27,65),(28,80),(29,91),(30,106),(31,93),(32,108),(33,95),(34,110),(35,97),(36,112),(37,99),(38,86),(39,101),(40,88),(41,103),(42,90),(43,105),(44,92),(45,107),(46,94),(47,109),(48,96),(49,111),(50,98),(51,85),(52,100),(53,87),(54,102),(55,89),(56,104)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | ··· | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | 2+ 1+4 | D4×D7 | D4×D7 | D4⋊8D14 |
| kernel | D28⋊19D4 | C22⋊D28 | D14⋊D4 | D28⋊C4 | C4⋊D28 | C4×C7⋊D4 | C23⋊D14 | C28⋊D4 | C7×C4⋊D4 | C22×D28 | C2×D4×D7 | D28 | C7⋊D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 3 | 4 | 4 | 3 | 6 | 3 | 3 | 9 | 1 | 6 | 6 | 6 |
Matrix representation of D28⋊19D4 ►in GL6(𝔽29)
| 28 | 11 | 0 | 0 | 0 | 0 |
| 13 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 22 | 0 | 0 |
| 0 | 0 | 25 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 19 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 18 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 0 | 13 | 28 |
| 28 | 11 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 18 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(29))| [28,13,0,0,0,0,11,1,0,0,0,0,0,0,0,25,0,0,0,0,22,18,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,16,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,19,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,18,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,13,0,0,0,0,11,28],[28,0,0,0,0,0,11,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,18,1] >;
D28⋊19D4 in GAP, Magma, Sage, TeX
D_{28}\rtimes_{19}D_4 % in TeX
G:=Group("D28:19D4"); // GroupNames label
G:=SmallGroup(448,1062);
// by ID
G=gap.SmallGroup(448,1062);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,1571,297,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^15,c*b*c^-1=d*b*d=a^14*b,d*c*d=c^-1>;
// generators/relations