direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×C7⋊D4, C24⋊6D14, C14.882+ 1+4, C7⋊5D42, C28⋊8(C2×D4), (C7×D4)⋊16D4, D14⋊9(C2×D4), C22⋊4(D4×D7), (C2×D4)⋊37D14, Dic7⋊5(C2×D4), (C22×D4)⋊7D7, C28⋊7D4⋊37C2, C28⋊2D4⋊39C2, C28⋊D4⋊28C2, (D4×Dic7)⋊38C2, (C22×C4)⋊27D14, C23⋊D14⋊29C2, D14⋊C4⋊35C22, (C2×D28)⋊38C22, (D4×C14)⋊56C22, C24⋊D7⋊11C2, C4⋊Dic7⋊44C22, Dic7⋊D4⋊40C2, (C2×C28).543C23, (C2×C14).296C24, Dic7⋊C4⋊73C22, (C23×C14)⋊13C22, (C22×C28)⋊23C22, (C4×Dic7)⋊41C22, C14.143(C22×D4), (C23×D7)⋊14C22, C2.91(D4⋊6D14), C23.D7⋊62C22, C23.205(C22×D7), C22.309(C23×D7), (C22×C14).230C23, (C2×Dic7).153C23, (C22×Dic7)⋊33C22, (C22×D7).240C23, (C2×D4×D7)⋊25C2, (D4×C2×C14)⋊5C2, C4⋊2(C2×C7⋊D4), (C2×C14)⋊8(C2×D4), C2.103(C2×D4×D7), (C4×C7⋊D4)⋊25C2, (C2×C4×D7)⋊30C22, C22⋊2(C2×C7⋊D4), (C22×C7⋊D4)⋊14C2, (C2×C7⋊D4)⋊45C22, C2.16(C22×C7⋊D4), (C2×C4).626(C22×D7), SmallGroup(448,1254)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C7⋊D4
G = < a,b,c,d,e | a4=b2=c7=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 2164 in 428 conjugacy classes, 123 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, D4, C23, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C4×D4, C22≀C2, C4⋊D4, C4⋊1D4, C22×D4, C22×D4, C4×D7, D28, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, D42, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, D14⋊C4, C23.D7, C23.D7, C2×C4×D7, C2×D28, D4×D7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, D4×C14, C23×D7, C23×C14, C4×C7⋊D4, C28⋊7D4, D4×Dic7, C23⋊D14, C28⋊2D4, Dic7⋊D4, C28⋊D4, C24⋊D7, C2×D4×D7, C22×C7⋊D4, D4×C2×C14, D4×C7⋊D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, C7⋊D4, C22×D7, D42, D4×D7, C2×C7⋊D4, C23×D7, C2×D4×D7, D4⋊6D14, C22×C7⋊D4, D4×C7⋊D4
►On 112 points
Generators in S112
(1 64 8 57)(2 65 9 58)(3 66 10 59)(4 67 11 60)(5 68 12 61)(6 69 13 62)(7 70 14 63)(15 78 22 71)(16 79 23 72)(17 80 24 73)(18 81 25 74)(19 82 26 75)(20 83 27 76)(21 84 28 77)(29 92 36 85)(30 93 37 86)(31 94 38 87)(32 95 39 88)(33 96 40 89)(34 97 41 90)(35 98 42 91)(43 106 50 99)(44 107 51 100)(45 108 52 101)(46 109 53 102)(47 110 54 103)(48 111 55 104)(49 112 56 105)
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 83)(14 84)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 99)(30 100)(31 101)(32 102)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 111)(42 112)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 97)(56 98)
(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 36 15 50)(2 42 16 56)(3 41 17 55)(4 40 18 54)(5 39 19 53)(6 38 20 52)(7 37 21 51)(8 29 22 43)(9 35 23 49)(10 34 24 48)(11 33 25 47)(12 32 26 46)(13 31 27 45)(14 30 28 44)(57 92 71 106)(58 98 72 112)(59 97 73 111)(60 96 74 110)(61 95 75 109)(62 94 76 108)(63 93 77 107)(64 85 78 99)(65 91 79 105)(66 90 80 104)(67 89 81 103)(68 88 82 102)(69 87 83 101)(70 86 84 100)
(1 8)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(15 22)(16 28)(17 27)(18 26)(19 25)(20 24)(21 23)(29 50)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 43)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(57 64)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(71 78)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(85 106)(86 112)(87 111)(88 110)(89 109)(90 108)(91 107)(92 99)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)
G:=sub<Sym(112)| (1,64,8,57)(2,65,9,58)(3,66,10,59)(4,67,11,60)(5,68,12,61)(6,69,13,62)(7,70,14,63)(15,78,22,71)(16,79,23,72)(17,80,24,73)(18,81,25,74)(19,82,26,75)(20,83,27,76)(21,84,28,77)(29,92,36,85)(30,93,37,86)(31,94,38,87)(32,95,39,88)(33,96,40,89)(34,97,41,90)(35,98,42,91)(43,106,50,99)(44,107,51,100)(45,108,52,101)(46,109,53,102)(47,110,54,103)(48,111,55,104)(49,112,56,105), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,99)(30,100)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,111)(42,112)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98), (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), (1,36,15,50)(2,42,16,56)(3,41,17,55)(4,40,18,54)(5,39,19,53)(6,38,20,52)(7,37,21,51)(8,29,22,43)(9,35,23,49)(10,34,24,48)(11,33,25,47)(12,32,26,46)(13,31,27,45)(14,30,28,44)(57,92,71,106)(58,98,72,112)(59,97,73,111)(60,96,74,110)(61,95,75,109)(62,94,76,108)(63,93,77,107)(64,85,78,99)(65,91,79,105)(66,90,80,104)(67,89,81,103)(68,88,82,102)(69,87,83,101)(70,86,84,100), (1,8)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(15,22)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(29,50)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,43)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(57,64)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(71,78)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(85,106)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,99)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)>;
G:=Group( (1,64,8,57)(2,65,9,58)(3,66,10,59)(4,67,11,60)(5,68,12,61)(6,69,13,62)(7,70,14,63)(15,78,22,71)(16,79,23,72)(17,80,24,73)(18,81,25,74)(19,82,26,75)(20,83,27,76)(21,84,28,77)(29,92,36,85)(30,93,37,86)(31,94,38,87)(32,95,39,88)(33,96,40,89)(34,97,41,90)(35,98,42,91)(43,106,50,99)(44,107,51,100)(45,108,52,101)(46,109,53,102)(47,110,54,103)(48,111,55,104)(49,112,56,105), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,99)(30,100)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,111)(42,112)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98), (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), (1,36,15,50)(2,42,16,56)(3,41,17,55)(4,40,18,54)(5,39,19,53)(6,38,20,52)(7,37,21,51)(8,29,22,43)(9,35,23,49)(10,34,24,48)(11,33,25,47)(12,32,26,46)(13,31,27,45)(14,30,28,44)(57,92,71,106)(58,98,72,112)(59,97,73,111)(60,96,74,110)(61,95,75,109)(62,94,76,108)(63,93,77,107)(64,85,78,99)(65,91,79,105)(66,90,80,104)(67,89,81,103)(68,88,82,102)(69,87,83,101)(70,86,84,100), (1,8)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(15,22)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(29,50)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,43)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(57,64)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(71,78)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(85,106)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,99)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100) );
G=PermutationGroup([[(1,64,8,57),(2,65,9,58),(3,66,10,59),(4,67,11,60),(5,68,12,61),(6,69,13,62),(7,70,14,63),(15,78,22,71),(16,79,23,72),(17,80,24,73),(18,81,25,74),(19,82,26,75),(20,83,27,76),(21,84,28,77),(29,92,36,85),(30,93,37,86),(31,94,38,87),(32,95,39,88),(33,96,40,89),(34,97,41,90),(35,98,42,91),(43,106,50,99),(44,107,51,100),(45,108,52,101),(46,109,53,102),(47,110,54,103),(48,111,55,104),(49,112,56,105)], [(1,71),(2,72),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,81),(12,82),(13,83),(14,84),(15,57),(16,58),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,99),(30,100),(31,101),(32,102),(33,103),(34,104),(35,105),(36,106),(37,107),(38,108),(39,109),(40,110),(41,111),(42,112),(43,85),(44,86),(45,87),(46,88),(47,89),(48,90),(49,91),(50,92),(51,93),(52,94),(53,95),(54,96),(55,97),(56,98)], [(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)], [(1,36,15,50),(2,42,16,56),(3,41,17,55),(4,40,18,54),(5,39,19,53),(6,38,20,52),(7,37,21,51),(8,29,22,43),(9,35,23,49),(10,34,24,48),(11,33,25,47),(12,32,26,46),(13,31,27,45),(14,30,28,44),(57,92,71,106),(58,98,72,112),(59,97,73,111),(60,96,74,110),(61,95,75,109),(62,94,76,108),(63,93,77,107),(64,85,78,99),(65,91,79,105),(66,90,80,104),(67,89,81,103),(68,88,82,102),(69,87,83,101),(70,86,84,100)], [(1,8),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(15,22),(16,28),(17,27),(18,26),(19,25),(20,24),(21,23),(29,50),(30,56),(31,55),(32,54),(33,53),(34,52),(35,51),(36,43),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(57,64),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(71,78),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(85,106),(86,112),(87,111),(88,110),(89,109),(90,108),(91,107),(92,99),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| 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 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
85 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | 2+ 1+4 | D4×D7 | D4⋊6D14 |
| kernel | D4×C7⋊D4 | C4×C7⋊D4 | C28⋊7D4 | D4×Dic7 | C23⋊D14 | C28⋊2D4 | Dic7⋊D4 | C28⋊D4 | C24⋊D7 | C2×D4×D7 | C22×C7⋊D4 | D4×C2×C14 | C7⋊D4 | C7×D4 | C22×D4 | C22×C4 | C2×D4 | C24 | D4 | C14 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 3 | 3 | 12 | 6 | 24 | 1 | 6 | 6 |
Matrix representation of D4×C7⋊D4 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 27 |
| 0 | 0 | 0 | 0 | 23 | 25 |
| 28 | 0 | 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 | 4 | 27 |
| 0 | 0 | 0 | 0 | 22 | 25 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 8 | 13 | 0 | 0 | 0 | 0 |
| 24 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 26 | 0 | 0 |
| 0 | 0 | 21 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 26 | 0 | 0 |
| 0 | 0 | 21 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,4,23,0,0,0,0,27,25],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,4,22,0,0,0,0,27,25],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,28,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,24,0,0,0,0,13,21,0,0,0,0,0,0,21,21,0,0,0,0,26,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,28,0,0,0,0,0,0,21,21,0,0,0,0,26,8,0,0,0,0,0,0,28,0,0,0,0,0,0,28] >;
D4×C7⋊D4 in GAP, Magma, Sage, TeX
D_4\times C_7\rtimes D_4
% in TeX
G:=Group("D4xC7:D4"); // GroupNames label
G:=SmallGroup(448,1254);
// by ID
G=gap.SmallGroup(448,1254);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,675,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^7=d^4=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations