direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D28, C42⋊13D14, C14.1022+ 1+4, C7⋊1D42, C4⋊4(D4×D7), (C7×D4)⋊9D4, C4⋊1(C2×D28), C28⋊1(C2×D4), C4⋊C4⋊46D14, D14⋊5(C2×D4), (C4×D4)⋊11D7, (D4×C28)⋊13C2, (C4×D28)⋊27C2, C28⋊7D4⋊7C2, C22⋊1(C2×D28), C22⋊D28⋊5C2, C4⋊D28⋊14C2, C28⋊4D4⋊11C2, (C4×C28)⋊18C22, D14⋊C4⋊4C22, C22⋊C4⋊45D14, (C22×D28)⋊8C2, (C22×C4)⋊11D14, (C2×D4).247D14, (C2×D28)⋊16C22, (C2×C14).93C24, C4⋊Dic7⋊58C22, (C22×C28)⋊9C22, C14.15(C22×D4), C2.17(C22×D28), (C23×D7)⋊5C22, (C2×C28).158C23, C2.14(D4⋊8D14), (D4×C14).256C22, (C2×Dic7).39C23, C23.171(C22×D7), C22.118(C23×D7), (C22×C14).163C23, (C22×D7).171C23, (C2×D4×D7)⋊3C2, C2.21(C2×D4×D7), (C2×C14)⋊1(C2×D4), (C2×C4×D7)⋊2C22, (C7×C4⋊C4)⋊58C22, (C2×C7⋊D4)⋊2C22, (C7×C22⋊C4)⋊49C22, (C2×C4).157(C22×D7), SmallGroup(448,1002)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×D28
G = < a,b,c,d | a4=b2=c28=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 2740 in 428 conjugacy classes, 123 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C4×D4, C4×D4, C22≀C2, C4⋊D4, C4⋊1D4, C22×D4, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, D42, C4⋊Dic7, D14⋊C4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, C2×D28, C2×D28, D4×D7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C4×D28, C28⋊4D4, C22⋊D28, C4⋊D28, C28⋊7D4, D4×C28, C22×D28, C2×D4×D7, D4×D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, D28, C22×D7, D42, C2×D28, D4×D7, C23×D7, C22×D28, C2×D4×D7, D4⋊8D14, D4×D28
►On 112 points
(1 74 35 91)(2 75 36 92)(3 76 37 93)(4 77 38 94)(5 78 39 95)(6 79 40 96)(7 80 41 97)(8 81 42 98)(9 82 43 99)(10 83 44 100)(11 84 45 101)(12 57 46 102)(13 58 47 103)(14 59 48 104)(15 60 49 105)(16 61 50 106)(17 62 51 107)(18 63 52 108)(19 64 53 109)(20 65 54 110)(21 66 55 111)(22 67 56 112)(23 68 29 85)(24 69 30 86)(25 70 31 87)(26 71 32 88)(27 72 33 89)(28 73 34 90)
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 97)(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 82)(30 83)(31 84)(32 57)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 65)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 81)
(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 41)(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(28 42)(57 86)(58 85)(59 112)(60 111)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 100)(72 99)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)(81 90)(82 89)(83 88)(84 87)
G:=sub<Sym(112)| (1,74,35,91)(2,75,36,92)(3,76,37,93)(4,77,38,94)(5,78,39,95)(6,79,40,96)(7,80,41,97)(8,81,42,98)(9,82,43,99)(10,83,44,100)(11,84,45,101)(12,57,46,102)(13,58,47,103)(14,59,48,104)(15,60,49,105)(16,61,50,106)(17,62,51,107)(18,63,52,108)(19,64,53,109)(20,65,54,110)(21,66,55,111)(22,67,56,112)(23,68,29,85)(24,69,30,86)(25,70,31,87)(26,71,32,88)(27,72,33,89)(28,73,34,90), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,82)(30,83)(31,84)(32,57)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,81), (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), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(57,86)(58,85)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)>;
G:=Group( (1,74,35,91)(2,75,36,92)(3,76,37,93)(4,77,38,94)(5,78,39,95)(6,79,40,96)(7,80,41,97)(8,81,42,98)(9,82,43,99)(10,83,44,100)(11,84,45,101)(12,57,46,102)(13,58,47,103)(14,59,48,104)(15,60,49,105)(16,61,50,106)(17,62,51,107)(18,63,52,108)(19,64,53,109)(20,65,54,110)(21,66,55,111)(22,67,56,112)(23,68,29,85)(24,69,30,86)(25,70,31,87)(26,71,32,88)(27,72,33,89)(28,73,34,90), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,82)(30,83)(31,84)(32,57)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,81), (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), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(57,86)(58,85)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87) );
G=PermutationGroup([[(1,74,35,91),(2,75,36,92),(3,76,37,93),(4,77,38,94),(5,78,39,95),(6,79,40,96),(7,80,41,97),(8,81,42,98),(9,82,43,99),(10,83,44,100),(11,84,45,101),(12,57,46,102),(13,58,47,103),(14,59,48,104),(15,60,49,105),(16,61,50,106),(17,62,51,107),(18,63,52,108),(19,64,53,109),(20,65,54,110),(21,66,55,111),(22,67,56,112),(23,68,29,85),(24,69,30,86),(25,70,31,87),(26,71,32,88),(27,72,33,89),(28,73,34,90)], [(1,105),(2,106),(3,107),(4,108),(5,109),(6,110),(7,111),(8,112),(9,85),(10,86),(11,87),(12,88),(13,89),(14,90),(15,91),(16,92),(17,93),(18,94),(19,95),(20,96),(21,97),(22,98),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(29,82),(30,83),(31,84),(32,57),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,65),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,73),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,81)], [(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)], [(1,41),(2,40),(3,39),(4,38),(5,37),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,29),(14,56),(15,55),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,48),(23,47),(24,46),(25,45),(26,44),(27,43),(28,42),(57,86),(58,85),(59,112),(60,111),(61,110),(62,109),(63,108),(64,107),(65,106),(66,105),(67,104),(68,103),(69,102),(70,101),(71,100),(72,99),(73,98),(74,97),(75,96),(76,95),(77,94),(78,93),(79,92),(80,91),(81,90),(82,89),(83,88),(84,87)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 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 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | D28 | 2+ 1+4 | D4×D7 | D4⋊8D14 |
| kernel | D4×D28 | C4×D28 | C28⋊4D4 | C22⋊D28 | C4⋊D28 | C28⋊7D4 | D4×C28 | C22×D28 | C2×D4×D7 | D28 | C7×D4 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 3 | 3 | 6 | 3 | 6 | 3 | 24 | 1 | 6 | 6 |
Matrix representation of D4×D28 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 16 |
| 0 | 0 | 10 | 19 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 10 | 16 |
| 0 | 0 | 21 | 19 |
| 12 | 5 | 0 | 0 |
| 24 | 27 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 21 | 3 | 0 | 0 |
| 8 | 8 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,10,10,0,0,16,19],[28,0,0,0,0,28,0,0,0,0,10,21,0,0,16,19],[12,24,0,0,5,27,0,0,0,0,28,0,0,0,0,28],[21,8,0,0,3,8,0,0,0,0,28,0,0,0,0,28] >;
D4×D28 in GAP, Magma, Sage, TeX
D_4\times D_{28} % in TeX
G:=Group("D4xD28"); // GroupNames label
G:=SmallGroup(448,1002);
// by ID
G=gap.SmallGroup(448,1002);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,675,80,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^28=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations