metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊20D4, C14.412+ (1+4), C4⋊C4⋊24D14, (C2×D4)⋊8D14, C4⋊D4⋊15D7, C7⋊6(D4⋊5D4), C4.110(D4×D7), D14⋊9(C4○D4), C28⋊2D4⋊21C2, C22⋊C4⋊28D14, D14.19(C2×D4), C28.229(C2×D4), (C22×C4)⋊19D14, D28⋊C4⋊21C2, C23⋊D14⋊11C2, D14⋊C4⋊18C22, D14⋊2Q8⋊22C2, (D4×C14)⋊14C22, C4⋊Dic7⋊32C22, C14.71(C22×D4), D14.D4⋊20C2, C28.17D4⋊17C2, (C2×C14).156C24, (C2×C28).595C23, Dic7⋊C4⋊65C22, (C22×C28)⋊22C22, (C4×Dic7)⋊23C22, C2.43(D4⋊6D14), C23.D7⋊24C22, (C2×Dic14)⋊62C22, (C2×D28).264C22, (C22×C14).23C23, (C2×Dic7).75C23, (C23×D7).48C22, C22.177(C23×D7), C23.113(C22×D7), (C22×D7).190C23, (C2×D4×D7)⋊13C2, C2.44(C2×D4×D7), (C4×C7⋊D4)⋊18C2, (D7×C22⋊C4)⋊6C2, C2.40(D7×C4○D4), (C2×C4×D7)⋊15C22, (C2×C4○D28)⋊22C2, (C7×C4⋊D4)⋊18C2, (C7×C4⋊C4)⋊13C22, C14.153(C2×C4○D4), (C2×C7⋊D4)⋊40C22, (C2×C4).39(C22×D7), (C7×C22⋊C4)⋊15C22, SmallGroup(448,1065)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1836 in 334 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×29], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×15], D4 [×18], Q8 [×2], C23, C23 [×2], C23 [×13], D7 [×6], C14 [×3], C14 [×3], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24 [×2], Dic7 [×5], C28 [×2], C28 [×3], D14 [×6], D14 [×14], C2×C14, C2×C14 [×9], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic14 [×2], C4×D7 [×8], D28 [×4], C2×Dic7 [×3], C2×Dic7 [×2], C7⋊D4 [×10], C2×C28 [×2], C2×C28 [×2], C2×C28 [×2], C7×D4 [×4], C22×D7, C22×D7 [×2], C22×D7 [×10], C22×C14, C22×C14 [×2], D4⋊5D4, C4×Dic7, Dic7⋊C4, C4⋊Dic7 [×2], D14⋊C4, D14⋊C4 [×4], C23.D7, C23.D7 [×4], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×C4×D7 [×4], C2×D28, C4○D28 [×4], D4×D7 [×4], C2×C7⋊D4, C2×C7⋊D4 [×4], C22×C28, D4×C14, D4×C14 [×2], C23×D7 [×2], D7×C22⋊C4 [×2], D14.D4 [×2], D28⋊C4, D14⋊2Q8, C4×C7⋊D4, C28.17D4, C23⋊D14 [×2], C28⋊2D4 [×2], C7×C4⋊D4, C2×C4○D28, C2×D4×D7, D28⋊20D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×2], C24, D14 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C22×D7 [×7], D4⋊5D4, D4×D7 [×2], C23×D7, C2×D4×D7, D4⋊6D14, D7×C4○D4, D28⋊20D4
Generators and relations
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, cac-1=dad=a15, bc=cb, 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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 28)(23 27)(24 26)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(54 56)(57 65)(58 64)(59 63)(60 62)(66 84)(67 83)(68 82)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)(85 109)(86 108)(87 107)(88 106)(89 105)(90 104)(91 103)(92 102)(93 101)(94 100)(95 99)(96 98)(110 112)
(1 31 101 65)(2 46 102 80)(3 33 103 67)(4 48 104 82)(5 35 105 69)(6 50 106 84)(7 37 107 71)(8 52 108 58)(9 39 109 73)(10 54 110 60)(11 41 111 75)(12 56 112 62)(13 43 85 77)(14 30 86 64)(15 45 87 79)(16 32 88 66)(17 47 89 81)(18 34 90 68)(19 49 91 83)(20 36 92 70)(21 51 93 57)(22 38 94 72)(23 53 95 59)(24 40 96 74)(25 55 97 61)(26 42 98 76)(27 29 99 63)(28 44 100 78)
(1 22)(2 9)(3 24)(4 11)(5 26)(6 13)(7 28)(8 15)(10 17)(12 19)(14 21)(16 23)(18 25)(20 27)(29 70)(30 57)(31 72)(32 59)(33 74)(34 61)(35 76)(36 63)(37 78)(38 65)(39 80)(40 67)(41 82)(42 69)(43 84)(44 71)(45 58)(46 73)(47 60)(48 75)(49 62)(50 77)(51 64)(52 79)(53 66)(54 81)(55 68)(56 83)(85 106)(86 93)(87 108)(88 95)(89 110)(90 97)(91 112)(92 99)(94 101)(96 103)(98 105)(100 107)(102 109)(104 111)
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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,65)(58,64)(59,63)(60,62)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(110,112), (1,31,101,65)(2,46,102,80)(3,33,103,67)(4,48,104,82)(5,35,105,69)(6,50,106,84)(7,37,107,71)(8,52,108,58)(9,39,109,73)(10,54,110,60)(11,41,111,75)(12,56,112,62)(13,43,85,77)(14,30,86,64)(15,45,87,79)(16,32,88,66)(17,47,89,81)(18,34,90,68)(19,49,91,83)(20,36,92,70)(21,51,93,57)(22,38,94,72)(23,53,95,59)(24,40,96,74)(25,55,97,61)(26,42,98,76)(27,29,99,63)(28,44,100,78), (1,22)(2,9)(3,24)(4,11)(5,26)(6,13)(7,28)(8,15)(10,17)(12,19)(14,21)(16,23)(18,25)(20,27)(29,70)(30,57)(31,72)(32,59)(33,74)(34,61)(35,76)(36,63)(37,78)(38,65)(39,80)(40,67)(41,82)(42,69)(43,84)(44,71)(45,58)(46,73)(47,60)(48,75)(49,62)(50,77)(51,64)(52,79)(53,66)(54,81)(55,68)(56,83)(85,106)(86,93)(87,108)(88,95)(89,110)(90,97)(91,112)(92,99)(94,101)(96,103)(98,105)(100,107)(102,109)(104,111)>;
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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,65)(58,64)(59,63)(60,62)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(110,112), (1,31,101,65)(2,46,102,80)(3,33,103,67)(4,48,104,82)(5,35,105,69)(6,50,106,84)(7,37,107,71)(8,52,108,58)(9,39,109,73)(10,54,110,60)(11,41,111,75)(12,56,112,62)(13,43,85,77)(14,30,86,64)(15,45,87,79)(16,32,88,66)(17,47,89,81)(18,34,90,68)(19,49,91,83)(20,36,92,70)(21,51,93,57)(22,38,94,72)(23,53,95,59)(24,40,96,74)(25,55,97,61)(26,42,98,76)(27,29,99,63)(28,44,100,78), (1,22)(2,9)(3,24)(4,11)(5,26)(6,13)(7,28)(8,15)(10,17)(12,19)(14,21)(16,23)(18,25)(20,27)(29,70)(30,57)(31,72)(32,59)(33,74)(34,61)(35,76)(36,63)(37,78)(38,65)(39,80)(40,67)(41,82)(42,69)(43,84)(44,71)(45,58)(46,73)(47,60)(48,75)(49,62)(50,77)(51,64)(52,79)(53,66)(54,81)(55,68)(56,83)(85,106)(86,93)(87,108)(88,95)(89,110)(90,97)(91,112)(92,99)(94,101)(96,103)(98,105)(100,107)(102,109)(104,111) );
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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,28),(23,27),(24,26),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(54,56),(57,65),(58,64),(59,63),(60,62),(66,84),(67,83),(68,82),(69,81),(70,80),(71,79),(72,78),(73,77),(74,76),(85,109),(86,108),(87,107),(88,106),(89,105),(90,104),(91,103),(92,102),(93,101),(94,100),(95,99),(96,98),(110,112)], [(1,31,101,65),(2,46,102,80),(3,33,103,67),(4,48,104,82),(5,35,105,69),(6,50,106,84),(7,37,107,71),(8,52,108,58),(9,39,109,73),(10,54,110,60),(11,41,111,75),(12,56,112,62),(13,43,85,77),(14,30,86,64),(15,45,87,79),(16,32,88,66),(17,47,89,81),(18,34,90,68),(19,49,91,83),(20,36,92,70),(21,51,93,57),(22,38,94,72),(23,53,95,59),(24,40,96,74),(25,55,97,61),(26,42,98,76),(27,29,99,63),(28,44,100,78)], [(1,22),(2,9),(3,24),(4,11),(5,26),(6,13),(7,28),(8,15),(10,17),(12,19),(14,21),(16,23),(18,25),(20,27),(29,70),(30,57),(31,72),(32,59),(33,74),(34,61),(35,76),(36,63),(37,78),(38,65),(39,80),(40,67),(41,82),(42,69),(43,84),(44,71),(45,58),(46,73),(47,60),(48,75),(49,62),(50,77),(51,64),(52,79),(53,66),(54,81),(55,68),(56,83),(85,106),(86,93),(87,108),(88,95),(89,110),(90,97),(91,112),(92,99),(94,101),(96,103),(98,105),(100,107),(102,109),(104,111)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 3 | 0 | 0 |
| 0 | 0 | 26 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 26 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [0,12,0,0,0,0,12,0,0,0,0,0,0,0,21,26,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,8,21,0,0,0,0,26,21,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,28,0],[0,12,0,0,0,0,17,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | ··· | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 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 | 4 | 4 | 4 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 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 | 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 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4×D7 | D4⋊6D14 | D7×C4○D4 |
| kernel | D28⋊20D4 | D7×C22⋊C4 | D14.D4 | D28⋊C4 | D14⋊2Q8 | C4×C7⋊D4 | C28.17D4 | C23⋊D14 | C28⋊2D4 | C7×C4⋊D4 | C2×C4○D28 | C2×D4×D7 | D28 | C4⋊D4 | D14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 3 | 4 | 6 | 3 | 3 | 9 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
D_{28}\rtimes_{20}D_4 % in TeX
G:=Group("D28:20D4"); // GroupNames label
G:=SmallGroup(448,1065);
// by ID
G=gap.SmallGroup(448,1065);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,1571,570,297,18822]);
// 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,b*c=c*b,d*b*d=a^14*b,d*c*d=c^-1>;
// generators/relations