metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊19D4, C14.1142+ (1+4), C7⋊3(D42), 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), C28⋊1D4⋊22C2, C28⋊D4⋊17C2, C22⋊C4⋊11D14, (C22×C4)⋊17D14, D28⋊C4⋊20C2, C22⋊D28⋊13C2, C23⋊D14⋊10C2, D14⋊D4⋊20C2, D14⋊C4⋊53C22, (C22×D28)⋊15C2, (C2×D28)⋊46C22, (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, C23.D7⋊51C22, C2.28(D4⋊8D14), (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
Subgroups: 2668 in 428 conjugacy classes, 115 normal (43 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×42], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×11], D4 [×34], C23, C23 [×2], C23 [×25], D7 [×8], C14 [×3], C14 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4, C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×29], C24 [×4], Dic7 [×2], Dic7 [×2], C28 [×2], C28 [×3], D14 [×6], D14 [×28], C2×C14, C2×C14 [×2], C2×C14 [×8], C4×D4 [×2], C22≀C2 [×4], C4⋊D4, C4⋊D4 [×3], C4⋊1D4, C22×D4 [×4], C4×D7 [×6], D28 [×4], D28 [×10], C2×Dic7 [×3], C7⋊D4 [×4], C7⋊D4 [×10], C2×C28 [×2], C2×C28 [×2], C2×C28 [×2], C7×D4 [×6], C22×D7, C22×D7 [×4], C22×D7 [×20], C22×C14, C22×C14 [×2], D42, C4×Dic7, Dic7⋊C4, D14⋊C4, D14⋊C4 [×4], C23.D7, C7×C22⋊C4 [×2], C7×C4⋊C4, C2×C4×D7, C2×C4×D7 [×2], C2×D28 [×2], C2×D28 [×4], C2×D28 [×4], D4×D7 [×12], C2×C7⋊D4, C2×C7⋊D4 [×6], C22×C28, D4×C14, D4×C14 [×2], C23×D7 [×4], C22⋊D28 [×2], D14⋊D4 [×2], D28⋊C4, C28⋊1D4, C4×C7⋊D4, C23⋊D14 [×2], C28⋊D4, C7×C4⋊D4, C22×D28, C2×D4×D7, C2×D4×D7 [×2], D28⋊19D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D7, C2×D4 [×12], C24, D14 [×7], C22×D4 [×2], 2+ (1+4), C22×D7 [×7], D42, D4×D7 [×4], C23×D7, C2×D4×D7 [×2], D4⋊8D14, D28⋊19D4
Generators and relations
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, cac-1=dad=a15, cbc-1=dbd=a14b, dcd=c-1 >
►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 65)(30 64)(31 63)(32 62)(33 61)(34 60)(35 59)(36 58)(37 57)(38 84)(39 83)(40 82)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)
(1 72 94 44)(2 59 95 31)(3 74 96 46)(4 61 97 33)(5 76 98 48)(6 63 99 35)(7 78 100 50)(8 65 101 37)(9 80 102 52)(10 67 103 39)(11 82 104 54)(12 69 105 41)(13 84 106 56)(14 71 107 43)(15 58 108 30)(16 73 109 45)(17 60 110 32)(18 75 111 47)(19 62 112 34)(20 77 85 49)(21 64 86 36)(22 79 87 51)(23 66 88 38)(24 81 89 53)(25 68 90 40)(26 83 91 55)(27 70 92 42)(28 57 93 29)
(1 58)(2 73)(3 60)(4 75)(5 62)(6 77)(7 64)(8 79)(9 66)(10 81)(11 68)(12 83)(13 70)(14 57)(15 72)(16 59)(17 74)(18 61)(19 76)(20 63)(21 78)(22 65)(23 80)(24 67)(25 82)(26 69)(27 84)(28 71)(29 107)(30 94)(31 109)(32 96)(33 111)(34 98)(35 85)(36 100)(37 87)(38 102)(39 89)(40 104)(41 91)(42 106)(43 93)(44 108)(45 95)(46 110)(47 97)(48 112)(49 99)(50 86)(51 101)(52 88)(53 103)(54 90)(55 105)(56 92)
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)(29,65)(30,64)(31,63)(32,62)(33,61)(34,60)(35,59)(36,58)(37,57)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66), (1,72,94,44)(2,59,95,31)(3,74,96,46)(4,61,97,33)(5,76,98,48)(6,63,99,35)(7,78,100,50)(8,65,101,37)(9,80,102,52)(10,67,103,39)(11,82,104,54)(12,69,105,41)(13,84,106,56)(14,71,107,43)(15,58,108,30)(16,73,109,45)(17,60,110,32)(18,75,111,47)(19,62,112,34)(20,77,85,49)(21,64,86,36)(22,79,87,51)(23,66,88,38)(24,81,89,53)(25,68,90,40)(26,83,91,55)(27,70,92,42)(28,57,93,29), (1,58)(2,73)(3,60)(4,75)(5,62)(6,77)(7,64)(8,79)(9,66)(10,81)(11,68)(12,83)(13,70)(14,57)(15,72)(16,59)(17,74)(18,61)(19,76)(20,63)(21,78)(22,65)(23,80)(24,67)(25,82)(26,69)(27,84)(28,71)(29,107)(30,94)(31,109)(32,96)(33,111)(34,98)(35,85)(36,100)(37,87)(38,102)(39,89)(40,104)(41,91)(42,106)(43,93)(44,108)(45,95)(46,110)(47,97)(48,112)(49,99)(50,86)(51,101)(52,88)(53,103)(54,90)(55,105)(56,92)>;
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)(29,65)(30,64)(31,63)(32,62)(33,61)(34,60)(35,59)(36,58)(37,57)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66), (1,72,94,44)(2,59,95,31)(3,74,96,46)(4,61,97,33)(5,76,98,48)(6,63,99,35)(7,78,100,50)(8,65,101,37)(9,80,102,52)(10,67,103,39)(11,82,104,54)(12,69,105,41)(13,84,106,56)(14,71,107,43)(15,58,108,30)(16,73,109,45)(17,60,110,32)(18,75,111,47)(19,62,112,34)(20,77,85,49)(21,64,86,36)(22,79,87,51)(23,66,88,38)(24,81,89,53)(25,68,90,40)(26,83,91,55)(27,70,92,42)(28,57,93,29), (1,58)(2,73)(3,60)(4,75)(5,62)(6,77)(7,64)(8,79)(9,66)(10,81)(11,68)(12,83)(13,70)(14,57)(15,72)(16,59)(17,74)(18,61)(19,76)(20,63)(21,78)(22,65)(23,80)(24,67)(25,82)(26,69)(27,84)(28,71)(29,107)(30,94)(31,109)(32,96)(33,111)(34,98)(35,85)(36,100)(37,87)(38,102)(39,89)(40,104)(41,91)(42,106)(43,93)(44,108)(45,95)(46,110)(47,97)(48,112)(49,99)(50,86)(51,101)(52,88)(53,103)(54,90)(55,105)(56,92) );
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),(29,65),(30,64),(31,63),(32,62),(33,61),(34,60),(35,59),(36,58),(37,57),(38,84),(39,83),(40,82),(41,81),(42,80),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(49,73),(50,72),(51,71),(52,70),(53,69),(54,68),(55,67),(56,66)], [(1,72,94,44),(2,59,95,31),(3,74,96,46),(4,61,97,33),(5,76,98,48),(6,63,99,35),(7,78,100,50),(8,65,101,37),(9,80,102,52),(10,67,103,39),(11,82,104,54),(12,69,105,41),(13,84,106,56),(14,71,107,43),(15,58,108,30),(16,73,109,45),(17,60,110,32),(18,75,111,47),(19,62,112,34),(20,77,85,49),(21,64,86,36),(22,79,87,51),(23,66,88,38),(24,81,89,53),(25,68,90,40),(26,83,91,55),(27,70,92,42),(28,57,93,29)], [(1,58),(2,73),(3,60),(4,75),(5,62),(6,77),(7,64),(8,79),(9,66),(10,81),(11,68),(12,83),(13,70),(14,57),(15,72),(16,59),(17,74),(18,61),(19,76),(20,63),(21,78),(22,65),(23,80),(24,67),(25,82),(26,69),(27,84),(28,71),(29,107),(30,94),(31,109),(32,96),(33,111),(34,98),(35,85),(36,100),(37,87),(38,102),(39,89),(40,104),(41,91),(42,106),(43,93),(44,108),(45,95),(46,110),(47,97),(48,112),(49,99),(50,86),(51,101),(52,88),(53,103),(54,90),(55,105),(56,92)])
Matrix representation ►G ⊆ 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] >;
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 | C28⋊1D4 | 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 |
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