metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.12D28, Q8.12D28, D56⋊12C22, C28.62C24, C56.11C23, 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, C4.28(C2×D28), C28.74(C2×D4), (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, C2.31(C22×D28), C14.29(C22×D4), (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
Generators and relations for D4.12D28
G = < a,b,c,d | a4=b2=d2=1, c28=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c27 >
Subgroups: 1692 in 268 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C8○D4, C2×D8, C4○D8, C8⋊C22, 2+ 1+4, C56, C56, Dic14, C4×D7, D28, D28, C7⋊D4, C2×C28, C7×D4, C7×Q8, C22×D7, D4○D8, C56⋊C2, D56, Dic28, C2×C56, C7×M4(2), C2×D28, C4○D28, D4×D7, Q8⋊2D7, C7×C4○D4, C2×D56, D56⋊7C2, C8⋊D14, C7×C8○D4, D4⋊8D14, D4.12D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, D28, C22×D7, D4○D8, C2×D28, C23×D7, C22×D28, D4.12D28
►On 112 points
Generators in S112
(1 73 29 101)(2 74 30 102)(3 75 31 103)(4 76 32 104)(5 77 33 105)(6 78 34 106)(7 79 35 107)(8 80 36 108)(9 81 37 109)(10 82 38 110)(11 83 39 111)(12 84 40 112)(13 85 41 57)(14 86 42 58)(15 87 43 59)(16 88 44 60)(17 89 45 61)(18 90 46 62)(19 91 47 63)(20 92 48 64)(21 93 49 65)(22 94 50 66)(23 95 51 67)(24 96 52 68)(25 97 53 69)(26 98 54 70)(27 99 55 71)(28 100 56 72)
(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 95)(58 94)(59 93)(60 92)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 105)
G:=sub<Sym(112)| (1,73,29,101)(2,74,30,102)(3,75,31,103)(4,76,32,104)(5,77,33,105)(6,78,34,106)(7,79,35,107)(8,80,36,108)(9,81,37,109)(10,82,38,110)(11,83,39,111)(12,84,40,112)(13,85,41,57)(14,86,42,58)(15,87,43,59)(16,88,44,60)(17,89,45,61)(18,90,46,62)(19,91,47,63)(20,92,48,64)(21,93,49,65)(22,94,50,66)(23,95,51,67)(24,96,52,68)(25,97,53,69)(26,98,54,70)(27,99,55,71)(28,100,56,72), (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)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)>;
G:=Group( (1,73,29,101)(2,74,30,102)(3,75,31,103)(4,76,32,104)(5,77,33,105)(6,78,34,106)(7,79,35,107)(8,80,36,108)(9,81,37,109)(10,82,38,110)(11,83,39,111)(12,84,40,112)(13,85,41,57)(14,86,42,58)(15,87,43,59)(16,88,44,60)(17,89,45,61)(18,90,46,62)(19,91,47,63)(20,92,48,64)(21,93,49,65)(22,94,50,66)(23,95,51,67)(24,96,52,68)(25,97,53,69)(26,98,54,70)(27,99,55,71)(28,100,56,72), (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)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105) );
G=PermutationGroup([[(1,73,29,101),(2,74,30,102),(3,75,31,103),(4,76,32,104),(5,77,33,105),(6,78,34,106),(7,79,35,107),(8,80,36,108),(9,81,37,109),(10,82,38,110),(11,83,39,111),(12,84,40,112),(13,85,41,57),(14,86,42,58),(15,87,43,59),(16,88,44,60),(17,89,45,61),(18,90,46,62),(19,91,47,63),(20,92,48,64),(21,93,49,65),(22,94,50,66),(23,95,51,67),(24,96,52,68),(25,97,53,69),(26,98,54,70),(27,99,55,71),(28,100,56,72)], [(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),(57,95),(58,94),(59,93),(60,92),(61,91),(62,90),(63,89),(64,88),(65,87),(66,86),(67,85),(68,84),(69,83),(70,82),(71,81),(72,80),(73,79),(74,78),(75,77),(96,112),(97,111),(98,110),(99,109),(100,108),(101,107),(102,106),(103,105)]])
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 |
Matrix representation of D4.12D28 ►in 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] >;
D4.12D28 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