metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.12D28, Q8.12D28, D56⋊12C22, C56.11C23, C28.62C24, M4(2)⋊21D14, D28.25C23, Dic28⋊21C22, Dic14.25C23, C8○D4⋊4D7, (C2×C8)⋊7D14, C7⋊1(D4○D8), (C2×D56)⋊15C2, (C7×D4).24D4, C28.74(C2×D4), C4.28(C2×D28), (C7×Q8).24D4, D4⋊8D14⋊4C2, C8⋊D14⋊12C2, (C2×C56)⋊10C22, C4○D4.39D14, C4○D28⋊2C22, D56⋊7C2⋊12C2, C4.59(C23×D7), C8.53(C22×D7), C22.4(C2×D28), (C2×D28)⋊31C22, C56⋊C2⋊12C22, C14.29(C22×D4), C2.31(C22×D28), (C2×C28).516C23, (C7×M4(2))⋊23C22, (C7×C8○D4)⋊4C2, (C2×C14).9(C2×D4), (C7×C4○D4).46C22, (C2×C4).227(C22×D7), SmallGroup(448,1205)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1692 in 268 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2 [×9], C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C7, C8, C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×18], Q8, Q8 [×2], C23 [×6], D7 [×6], C14, C14 [×3], C2×C8 [×3], M4(2) [×3], D8 [×9], SD16 [×6], Q16, C2×D4 [×12], C4○D4, C4○D4 [×8], Dic7 [×2], C28, C28 [×3], D14 [×12], C2×C14 [×3], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22 [×6], 2+ (1+4) [×2], C56, C56 [×3], Dic14 [×2], C4×D7 [×6], D28 [×6], D28 [×6], C7⋊D4 [×6], C2×C28 [×3], C7×D4 [×3], C7×Q8, C22×D7 [×6], D4○D8, C56⋊C2 [×6], D56 [×9], Dic28, C2×C56 [×3], C7×M4(2) [×3], C2×D28 [×6], C4○D28 [×6], D4×D7 [×6], Q8⋊2D7 [×2], C7×C4○D4, C2×D56 [×3], D56⋊7C2 [×3], C8⋊D14 [×6], C7×C8○D4, D4⋊8D14 [×2], D4.12D28
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, D28 [×4], C22×D7 [×7], D4○D8, C2×D28 [×6], C23×D7, C22×D28, D4.12D28
Generators and relations
G = < a,b,c,d | a4=b2=d2=1, c28=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c27 >
►On 112 points
(1 103 29 75)(2 104 30 76)(3 105 31 77)(4 106 32 78)(5 107 33 79)(6 108 34 80)(7 109 35 81)(8 110 36 82)(9 111 37 83)(10 112 38 84)(11 57 39 85)(12 58 40 86)(13 59 41 87)(14 60 42 88)(15 61 43 89)(16 62 44 90)(17 63 45 91)(18 64 46 92)(19 65 47 93)(20 66 48 94)(21 67 49 95)(22 68 50 96)(23 69 51 97)(24 70 52 98)(25 71 53 99)(26 72 54 100)(27 73 55 101)(28 74 56 102)
(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(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 7)(2 6)(3 5)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(57 99)(58 98)(59 97)(60 96)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)
G:=sub<Sym(112)| (1,103,29,75)(2,104,30,76)(3,105,31,77)(4,106,32,78)(5,107,33,79)(6,108,34,80)(7,109,35,81)(8,110,36,82)(9,111,37,83)(10,112,38,84)(11,57,39,85)(12,58,40,86)(13,59,41,87)(14,60,42,88)(15,61,43,89)(16,62,44,90)(17,63,45,91)(18,64,46,92)(19,65,47,93)(20,66,48,94)(21,67,49,95)(22,68,50,96)(23,69,51,97)(24,70,52,98)(25,71,53,99)(26,72,54,100)(27,73,55,101)(28,74,56,102), (57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (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), (1,7)(2,6)(3,5)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)>;
G:=Group( (1,103,29,75)(2,104,30,76)(3,105,31,77)(4,106,32,78)(5,107,33,79)(6,108,34,80)(7,109,35,81)(8,110,36,82)(9,111,37,83)(10,112,38,84)(11,57,39,85)(12,58,40,86)(13,59,41,87)(14,60,42,88)(15,61,43,89)(16,62,44,90)(17,63,45,91)(18,64,46,92)(19,65,47,93)(20,66,48,94)(21,67,49,95)(22,68,50,96)(23,69,51,97)(24,70,52,98)(25,71,53,99)(26,72,54,100)(27,73,55,101)(28,74,56,102), (57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (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), (1,7)(2,6)(3,5)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107) );
G=PermutationGroup([(1,103,29,75),(2,104,30,76),(3,105,31,77),(4,106,32,78),(5,107,33,79),(6,108,34,80),(7,109,35,81),(8,110,36,82),(9,111,37,83),(10,112,38,84),(11,57,39,85),(12,58,40,86),(13,59,41,87),(14,60,42,88),(15,61,43,89),(16,62,44,90),(17,63,45,91),(18,64,46,92),(19,65,47,93),(20,66,48,94),(21,67,49,95),(22,68,50,96),(23,69,51,97),(24,70,52,98),(25,71,53,99),(26,72,54,100),(27,73,55,101),(28,74,56,102)], [(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(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)], [(1,7),(2,6),(3,5),(8,56),(9,55),(10,54),(11,53),(12,52),(13,51),(14,50),(15,49),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(57,99),(58,98),(59,97),(60,96),(61,95),(62,94),(63,93),(64,92),(65,91),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(100,112),(101,111),(102,110),(103,109),(104,108),(105,107)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 49 | 61 | 0 | 0 |
| 52 | 44 | 0 | 0 |
| 0 | 0 | 49 | 61 |
| 0 | 0 | 52 | 44 |
| 90 | 77 | 0 | 0 |
| 90 | 23 | 0 | 0 |
| 0 | 0 | 90 | 77 |
| 0 | 0 | 90 | 23 |
G:=sub<GL(4,GF(113))| [0,0,1,0,0,0,0,1,112,0,0,0,0,112,0,0],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[49,52,0,0,61,44,0,0,0,0,49,52,0,0,61,44],[90,90,0,0,77,23,0,0,0,0,90,90,0,0,77,23] >;
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | ··· | 14L | 28A | ··· | 28F | 28G | ··· | 28O | 56A | ··· | 56L | 56M | ··· | 56AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | D28 | D28 | D4○D8 | D4.12D28 |
| kernel | D4.12D28 | C2×D56 | D56⋊7C2 | C8⋊D14 | C7×C8○D4 | D4⋊8D14 | C7×D4 | C7×Q8 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C7 | C1 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 3 | 1 | 3 | 9 | 9 | 3 | 18 | 6 | 2 | 12 |
In GAP, Magma, Sage, TeX
D_4._{12}D_{28} % in TeX
G:=Group("D4.12D28"); // GroupNames label
G:=SmallGroup(448,1205);
// by ID
G=gap.SmallGroup(448,1205);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,675,192,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^28=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c^27>;
// generators/relations