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×C28).595C23, (C2×C14).156C24, Dic7⋊C4⋊65C22, (C22×C28)⋊22C22, (C4×Dic7)⋊23C22, C23.D7⋊24C22, C2.43(D4⋊6D14), (C2×Dic14)⋊62C22, (C2×D28).264C22, (C22×C14).23C23, (C2×Dic7).75C23, (C23×D7).48C22, C23.113(C22×D7), C22.177(C23×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
Generators and relations for D28⋊20D4
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, cac-1=dad=a15, bc=cb, dbd=a14b, dcd=c-1 >
Subgroups: 1836 in 334 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, 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, C2×Q8, C4○D4, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C22×C14, D4⋊5D4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, D14⋊C4, C23.D7, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×D28, C4○D28, D4×D7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C23×D7, 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⋊20D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, 2+ 1+4, C22×D7, D4⋊5D4, D4×D7, C23×D7, C2×D4×D7, D4⋊6D14, D7×C4○D4, D28⋊20D4
►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 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(46 56)(47 55)(48 54)(49 53)(50 52)(58 84)(59 83)(60 82)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(85 107)(86 106)(87 105)(88 104)(89 103)(90 102)(91 101)(92 100)(93 99)(94 98)(95 97)(108 112)(109 111)
(1 75 100 41)(2 62 101 56)(3 77 102 43)(4 64 103 30)(5 79 104 45)(6 66 105 32)(7 81 106 47)(8 68 107 34)(9 83 108 49)(10 70 109 36)(11 57 110 51)(12 72 111 38)(13 59 112 53)(14 74 85 40)(15 61 86 55)(16 76 87 42)(17 63 88 29)(18 78 89 44)(19 65 90 31)(20 80 91 46)(21 67 92 33)(22 82 93 48)(23 69 94 35)(24 84 95 50)(25 71 96 37)(26 58 97 52)(27 73 98 39)(28 60 99 54)
(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 92)(86 107)(87 94)(88 109)(89 96)(90 111)(91 98)(93 100)(95 102)(97 104)(99 106)(101 108)(103 110)(105 112)
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)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,56)(47,55)(48,54)(49,53)(50,52)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(108,112)(109,111), (1,75,100,41)(2,62,101,56)(3,77,102,43)(4,64,103,30)(5,79,104,45)(6,66,105,32)(7,81,106,47)(8,68,107,34)(9,83,108,49)(10,70,109,36)(11,57,110,51)(12,72,111,38)(13,59,112,53)(14,74,85,40)(15,61,86,55)(16,76,87,42)(17,63,88,29)(18,78,89,44)(19,65,90,31)(20,80,91,46)(21,67,92,33)(22,82,93,48)(23,69,94,35)(24,84,95,50)(25,71,96,37)(26,58,97,52)(27,73,98,39)(28,60,99,54), (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)(85,92)(86,107)(87,94)(88,109)(89,96)(90,111)(91,98)(93,100)(95,102)(97,104)(99,106)(101,108)(103,110)(105,112)>;
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)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,56)(47,55)(48,54)(49,53)(50,52)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(108,112)(109,111), (1,75,100,41)(2,62,101,56)(3,77,102,43)(4,64,103,30)(5,79,104,45)(6,66,105,32)(7,81,106,47)(8,68,107,34)(9,83,108,49)(10,70,109,36)(11,57,110,51)(12,72,111,38)(13,59,112,53)(14,74,85,40)(15,61,86,55)(16,76,87,42)(17,63,88,29)(18,78,89,44)(19,65,90,31)(20,80,91,46)(21,67,92,33)(22,82,93,48)(23,69,94,35)(24,84,95,50)(25,71,96,37)(26,58,97,52)(27,73,98,39)(28,60,99,54), (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)(85,92)(86,107)(87,94)(88,109)(89,96)(90,111)(91,98)(93,100)(95,102)(97,104)(99,106)(101,108)(103,110)(105,112) );
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),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38),(46,56),(47,55),(48,54),(49,53),(50,52),(58,84),(59,83),(60,82),(61,81),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(85,107),(86,106),(87,105),(88,104),(89,103),(90,102),(91,101),(92,100),(93,99),(94,98),(95,97),(108,112),(109,111)], [(1,75,100,41),(2,62,101,56),(3,77,102,43),(4,64,103,30),(5,79,104,45),(6,66,105,32),(7,81,106,47),(8,68,107,34),(9,83,108,49),(10,70,109,36),(11,57,110,51),(12,72,111,38),(13,59,112,53),(14,74,85,40),(15,61,86,55),(16,76,87,42),(17,63,88,29),(18,78,89,44),(19,65,90,31),(20,80,91,46),(21,67,92,33),(22,82,93,48),(23,69,94,35),(24,84,95,50),(25,71,96,37),(26,58,97,52),(27,73,98,39),(28,60,99,54)], [(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),(85,92),(86,107),(87,94),(88,109),(89,96),(90,111),(91,98),(93,100),(95,102),(97,104),(99,106),(101,108),(103,110),(105,112)]])
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 |
Matrix representation of D28⋊20D4 ►in 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] >;
D28⋊20D4 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