direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D28, C42⋊13D14, C14.1022+ (1+4), C7⋊1(D42), 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, (C4×D28)⋊27C2, (D4×C28)⋊13C2, C28⋊7D4⋊7C2, C22⋊1(C2×D28), C22⋊D28⋊5C2, C28⋊1D4⋊14C2, C4⋊D28⋊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, C22.118(C23×D7), C23.171(C22×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
Subgroups: 2740 in 428 conjugacy classes, 123 normal (29 characteristic)
C1, C2 [×3], C2 [×12], C4 [×4], C4 [×5], C22, C22 [×4], C22 [×40], C7, C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×4], D4 [×30], C23 [×2], C23 [×26], D7 [×8], C14 [×3], C14 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4, C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×31], C24 [×4], Dic7 [×2], C28 [×4], C28 [×3], D14 [×4], D14 [×32], C2×C14, C2×C14 [×4], C2×C14 [×4], C4×D4, C4×D4, C22≀C2 [×4], C4⋊D4 [×4], C4⋊1D4, C22×D4 [×4], C4×D7 [×4], D28 [×4], D28 [×18], C2×Dic7 [×2], C7⋊D4 [×8], C2×C28 [×3], C2×C28 [×2], C2×C28 [×4], C7×D4 [×4], C22×D7 [×6], C22×D7 [×20], C22×C14 [×2], D42, C4⋊Dic7, D14⋊C4 [×6], C4×C28, C7×C22⋊C4 [×2], C7×C4⋊C4, C2×C4×D7 [×2], C2×D28, C2×D28 [×10], C2×D28 [×8], D4×D7 [×8], C2×C7⋊D4 [×4], C22×C28 [×2], D4×C14, C23×D7 [×4], C4×D28, C4⋊D28, C22⋊D28 [×4], C28⋊1D4 [×2], C28⋊7D4 [×2], D4×C28, C22×D28 [×2], C2×D4×D7 [×2], D4×D28
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D7, C2×D4 [×12], C24, D14 [×7], C22×D4 [×2], 2+ (1+4), D28 [×4], C22×D7 [×7], D42, C2×D28 [×6], D4×D7 [×2], C23×D7, C22×D28, C2×D4×D7, D4⋊8D14, D4×D28
Generators and relations
G = < a,b,c,d | a4=b2=c28=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 112 points
(1 82 32 91)(2 83 33 92)(3 84 34 93)(4 57 35 94)(5 58 36 95)(6 59 37 96)(7 60 38 97)(8 61 39 98)(9 62 40 99)(10 63 41 100)(11 64 42 101)(12 65 43 102)(13 66 44 103)(14 67 45 104)(15 68 46 105)(16 69 47 106)(17 70 48 107)(18 71 49 108)(19 72 50 109)(20 73 51 110)(21 74 52 111)(22 75 53 112)(23 76 54 85)(24 77 55 86)(25 78 56 87)(26 79 29 88)(27 80 30 89)(28 81 31 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 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(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 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 56)(12 55)(13 54)(14 53)(15 52)(16 51)(17 50)(18 49)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(27 40)(28 39)(57 94)(58 93)(59 92)(60 91)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 112)(68 111)(69 110)(70 109)(71 108)(72 107)(73 106)(74 105)(75 104)(76 103)(77 102)(78 101)(79 100)(80 99)(81 98)(82 97)(83 96)(84 95)
G:=sub<Sym(112)| (1,82,32,91)(2,83,33,92)(3,84,34,93)(4,57,35,94)(5,58,36,95)(6,59,37,96)(7,60,38,97)(8,61,39,98)(9,62,40,99)(10,63,41,100)(11,64,42,101)(12,65,43,102)(13,66,44,103)(14,67,45,104)(15,68,46,105)(16,69,47,106)(17,70,48,107)(18,71,49,108)(19,72,50,109)(20,73,51,110)(21,74,52,111)(22,75,53,112)(23,76,54,85)(24,77,55,86)(25,78,56,87)(26,79,29,88)(27,80,30,89)(28,81,31,90), (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)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,112)(68,111)(69,110)(70,109)(71,108)(72,107)(73,106)(74,105)(75,104)(76,103)(77,102)(78,101)(79,100)(80,99)(81,98)(82,97)(83,96)(84,95)>;
G:=Group( (1,82,32,91)(2,83,33,92)(3,84,34,93)(4,57,35,94)(5,58,36,95)(6,59,37,96)(7,60,38,97)(8,61,39,98)(9,62,40,99)(10,63,41,100)(11,64,42,101)(12,65,43,102)(13,66,44,103)(14,67,45,104)(15,68,46,105)(16,69,47,106)(17,70,48,107)(18,71,49,108)(19,72,50,109)(20,73,51,110)(21,74,52,111)(22,75,53,112)(23,76,54,85)(24,77,55,86)(25,78,56,87)(26,79,29,88)(27,80,30,89)(28,81,31,90), (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)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,112)(68,111)(69,110)(70,109)(71,108)(72,107)(73,106)(74,105)(75,104)(76,103)(77,102)(78,101)(79,100)(80,99)(81,98)(82,97)(83,96)(84,95) );
G=PermutationGroup([(1,82,32,91),(2,83,33,92),(3,84,34,93),(4,57,35,94),(5,58,36,95),(6,59,37,96),(7,60,38,97),(8,61,39,98),(9,62,40,99),(10,63,41,100),(11,64,42,101),(12,65,43,102),(13,66,44,103),(14,67,45,104),(15,68,46,105),(16,69,47,106),(17,70,48,107),(18,71,49,108),(19,72,50,109),(20,73,51,110),(21,74,52,111),(22,75,53,112),(23,76,54,85),(24,77,55,86),(25,78,56,87),(26,79,29,88),(27,80,30,89),(28,81,31,90)], [(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),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,79),(44,80),(45,81),(46,82),(47,83),(48,84),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)], [(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)], [(1,38),(2,37),(3,36),(4,35),(5,34),(6,33),(7,32),(8,31),(9,30),(10,29),(11,56),(12,55),(13,54),(14,53),(15,52),(16,51),(17,50),(18,49),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(25,42),(26,41),(27,40),(28,39),(57,94),(58,93),(59,92),(60,91),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,112),(68,111),(69,110),(70,109),(71,108),(72,107),(73,106),(74,105),(75,104),(76,103),(77,102),(78,101),(79,100),(80,99),(81,98),(82,97),(83,96),(84,95)])
Matrix representation ►G ⊆ 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] >;
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 | C4⋊D28 | C22⋊D28 | C28⋊1D4 | 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 |
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