metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.38D4, Dic14.38D4, M4(2).14D14, C8⋊C22⋊3D7, C4○D4.6D14, (C7×D4).13D4, C4.105(D4×D7), (C7×Q8).13D4, (C2×D4).81D14, C28.197(C2×D4), C7⋊4(D4.8D4), (C2×Dic7).5D4, D28⋊4C4⋊10C2, C22.36(D4×D7), C14.64C22≀C2, D4⋊2Dic7⋊7C2, D4.D14⋊6C2, C28.17D4⋊7C2, D4.10(C7⋊D4), (C2×C28).16C23, Q8.10(C7⋊D4), D4.10D14⋊2C2, C4.12D28⋊10C2, C4○D28.24C22, C2.32(C23⋊D14), (D4×C14).106C22, (C4×Dic7).58C22, C4.Dic7.26C22, (C7×M4(2)).24C22, (C2×Dic14).135C22, (C7×C8⋊C22)⋊7C2, C4.53(C2×C7⋊D4), (C2×C14).35(C2×D4), (C2×C4).16(C22×D7), (C7×C4○D4).14C22, SmallGroup(448,735)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.38D4
G = < a,b,c,d | a28=b2=1, c4=d2=a14, bab=a-1, cac-1=a15, ad=da, cbc-1=a7b, bd=db, dcd-1=c3 >
Subgroups: 748 in 146 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C42, C22⋊C4, M4(2), M4(2), D8, SD16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, C8⋊C22, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×C14, D4.8D4, C4.Dic7, C4×Dic7, D4⋊D7, D4.D7, C23.D7, C7×M4(2), C7×D8, C7×SD16, C2×Dic14, C2×Dic14, C4○D28, C4○D28, D4⋊2D7, Q8×D7, D4×C14, C7×C4○D4, C4.12D28, D28⋊4C4, D4⋊2Dic7, D4.D14, C28.17D4, C7×C8⋊C22, D4.10D14, D28.38D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, D4.8D4, D4×D7, C2×C7⋊D4, C23⋊D14, D28.38D4
►On 112 points
Generators in S112
(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 95)(2 94)(3 93)(4 92)(5 91)(6 90)(7 89)(8 88)(9 87)(10 86)(11 85)(12 112)(13 111)(14 110)(15 109)(16 108)(17 107)(18 106)(19 105)(20 104)(21 103)(22 102)(23 101)(24 100)(25 99)(26 98)(27 97)(28 96)(29 74)(30 73)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)(41 62)(42 61)(43 60)(44 59)(45 58)(46 57)(47 84)(48 83)(49 82)(50 81)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)
(1 89 22 96 15 103 8 110)(2 104 23 111 16 90 9 97)(3 91 24 98 17 105 10 112)(4 106 25 85 18 92 11 99)(5 93 26 100 19 107 12 86)(6 108 27 87 20 94 13 101)(7 95 28 102 21 109 14 88)(29 66 50 73 43 80 36 59)(30 81 51 60 44 67 37 74)(31 68 52 75 45 82 38 61)(32 83 53 62 46 69 39 76)(33 70 54 77 47 84 40 63)(34 57 55 64 48 71 41 78)(35 72 56 79 49 58 42 65)
(1 74 15 60)(2 75 16 61)(3 76 17 62)(4 77 18 63)(5 78 19 64)(6 79 20 65)(7 80 21 66)(8 81 22 67)(9 82 23 68)(10 83 24 69)(11 84 25 70)(12 57 26 71)(13 58 27 72)(14 59 28 73)(29 109 43 95)(30 110 44 96)(31 111 45 97)(32 112 46 98)(33 85 47 99)(34 86 48 100)(35 87 49 101)(36 88 50 102)(37 89 51 103)(38 90 52 104)(39 91 53 105)(40 92 54 106)(41 93 55 107)(42 94 56 108)
G:=sub<Sym(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), (1,95)(2,94)(3,93)(4,92)(5,91)(6,90)(7,89)(8,88)(9,87)(10,86)(11,85)(12,112)(13,111)(14,110)(15,109)(16,108)(17,107)(18,106)(19,105)(20,104)(21,103)(22,102)(23,101)(24,100)(25,99)(26,98)(27,97)(28,96)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75), (1,89,22,96,15,103,8,110)(2,104,23,111,16,90,9,97)(3,91,24,98,17,105,10,112)(4,106,25,85,18,92,11,99)(5,93,26,100,19,107,12,86)(6,108,27,87,20,94,13,101)(7,95,28,102,21,109,14,88)(29,66,50,73,43,80,36,59)(30,81,51,60,44,67,37,74)(31,68,52,75,45,82,38,61)(32,83,53,62,46,69,39,76)(33,70,54,77,47,84,40,63)(34,57,55,64,48,71,41,78)(35,72,56,79,49,58,42,65), (1,74,15,60)(2,75,16,61)(3,76,17,62)(4,77,18,63)(5,78,19,64)(6,79,20,65)(7,80,21,66)(8,81,22,67)(9,82,23,68)(10,83,24,69)(11,84,25,70)(12,57,26,71)(13,58,27,72)(14,59,28,73)(29,109,43,95)(30,110,44,96)(31,111,45,97)(32,112,46,98)(33,85,47,99)(34,86,48,100)(35,87,49,101)(36,88,50,102)(37,89,51,103)(38,90,52,104)(39,91,53,105)(40,92,54,106)(41,93,55,107)(42,94,56,108)>;
G:=Group( (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), (1,95)(2,94)(3,93)(4,92)(5,91)(6,90)(7,89)(8,88)(9,87)(10,86)(11,85)(12,112)(13,111)(14,110)(15,109)(16,108)(17,107)(18,106)(19,105)(20,104)(21,103)(22,102)(23,101)(24,100)(25,99)(26,98)(27,97)(28,96)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75), (1,89,22,96,15,103,8,110)(2,104,23,111,16,90,9,97)(3,91,24,98,17,105,10,112)(4,106,25,85,18,92,11,99)(5,93,26,100,19,107,12,86)(6,108,27,87,20,94,13,101)(7,95,28,102,21,109,14,88)(29,66,50,73,43,80,36,59)(30,81,51,60,44,67,37,74)(31,68,52,75,45,82,38,61)(32,83,53,62,46,69,39,76)(33,70,54,77,47,84,40,63)(34,57,55,64,48,71,41,78)(35,72,56,79,49,58,42,65), (1,74,15,60)(2,75,16,61)(3,76,17,62)(4,77,18,63)(5,78,19,64)(6,79,20,65)(7,80,21,66)(8,81,22,67)(9,82,23,68)(10,83,24,69)(11,84,25,70)(12,57,26,71)(13,58,27,72)(14,59,28,73)(29,109,43,95)(30,110,44,96)(31,111,45,97)(32,112,46,98)(33,85,47,99)(34,86,48,100)(35,87,49,101)(36,88,50,102)(37,89,51,103)(38,90,52,104)(39,91,53,105)(40,92,54,106)(41,93,55,107)(42,94,56,108) );
G=PermutationGroup([[(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)], [(1,95),(2,94),(3,93),(4,92),(5,91),(6,90),(7,89),(8,88),(9,87),(10,86),(11,85),(12,112),(13,111),(14,110),(15,109),(16,108),(17,107),(18,106),(19,105),(20,104),(21,103),(22,102),(23,101),(24,100),(25,99),(26,98),(27,97),(28,96),(29,74),(30,73),(31,72),(32,71),(33,70),(34,69),(35,68),(36,67),(37,66),(38,65),(39,64),(40,63),(41,62),(42,61),(43,60),(44,59),(45,58),(46,57),(47,84),(48,83),(49,82),(50,81),(51,80),(52,79),(53,78),(54,77),(55,76),(56,75)], [(1,89,22,96,15,103,8,110),(2,104,23,111,16,90,9,97),(3,91,24,98,17,105,10,112),(4,106,25,85,18,92,11,99),(5,93,26,100,19,107,12,86),(6,108,27,87,20,94,13,101),(7,95,28,102,21,109,14,88),(29,66,50,73,43,80,36,59),(30,81,51,60,44,67,37,74),(31,68,52,75,45,82,38,61),(32,83,53,62,46,69,39,76),(33,70,54,77,47,84,40,63),(34,57,55,64,48,71,41,78),(35,72,56,79,49,58,42,65)], [(1,74,15,60),(2,75,16,61),(3,76,17,62),(4,77,18,63),(5,78,19,64),(6,79,20,65),(7,80,21,66),(8,81,22,67),(9,82,23,68),(10,83,24,69),(11,84,25,70),(12,57,26,71),(13,58,27,72),(14,59,28,73),(29,109,43,95),(30,110,44,96),(31,111,45,97),(32,112,46,98),(33,85,47,99),(34,86,48,100),(35,87,49,101),(36,88,50,102),(37,89,51,103),(38,90,52,104),(39,91,53,105),(40,92,54,106),(41,93,55,107),(42,94,56,108)]])
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | ··· | 4H | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14O | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 8 | 28 | 2 | 2 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 8 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | D4.8D4 | D4×D7 | D4×D7 | D28.38D4 |
| kernel | D28.38D4 | C4.12D28 | D28⋊4C4 | D4⋊2Dic7 | D4.D14 | C28.17D4 | C7×C8⋊C22 | D4.10D14 | Dic14 | D28 | C2×Dic7 | C7×D4 | C7×Q8 | C8⋊C22 | M4(2) | C2×D4 | C4○D4 | D4 | Q8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | 6 | 2 | 3 | 3 | 3 |
Matrix representation of D28.38D4 ►in GL8(𝔽113)
| 33 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 104 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 98 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 56 | 15 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 | 0 | 98 |
| 41 | 15 | 43 | 0 | 0 | 0 | 0 | 0 |
| 45 | 72 | 106 | 70 | 0 | 0 | 0 | 0 |
| 74 | 0 | 14 | 15 | 0 | 0 | 0 | 0 |
| 30 | 39 | 31 | 99 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 31 | 0 | 8 |
| 0 | 0 | 0 | 0 | 111 | 30 | 8 | 0 |
| 0 | 0 | 0 | 0 | 39 | 79 | 83 | 2 |
| 0 | 0 | 0 | 0 | 15 | 17 | 82 | 100 |
| 63 | 13 | 0 | 65 | 0 | 0 | 0 | 0 |
| 100 | 67 | 55 | 63 | 0 | 0 | 0 | 0 |
| 104 | 56 | 36 | 100 | 0 | 0 | 0 | 0 |
| 58 | 87 | 102 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 100 | 0 | 106 |
| 0 | 0 | 0 | 0 | 30 | 2 | 106 | 0 |
| 0 | 0 | 0 | 0 | 93 | 60 | 111 | 83 |
| 0 | 0 | 0 | 0 | 31 | 84 | 13 | 82 |
| 101 | 13 | 0 | 103 | 0 | 0 | 0 | 0 |
| 100 | 105 | 35 | 112 | 0 | 0 | 0 | 0 |
| 90 | 83 | 111 | 100 | 0 | 0 | 0 | 0 |
| 20 | 87 | 102 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 2 | 106 | 0 |
| 0 | 0 | 0 | 0 | 31 | 100 | 0 | 106 |
| 0 | 0 | 0 | 0 | 73 | 21 | 83 | 111 |
| 0 | 0 | 0 | 0 | 43 | 17 | 82 | 13 |
G:=sub<GL(8,GF(113))| [33,104,0,0,0,0,0,0,9,1,0,0,0,0,0,0,0,0,34,25,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,15,0,0,36,0,0,0,0,0,98,56,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,98],[41,45,74,30,0,0,0,0,15,72,0,39,0,0,0,0,43,106,14,31,0,0,0,0,0,70,15,99,0,0,0,0,0,0,0,0,13,111,39,15,0,0,0,0,31,30,79,17,0,0,0,0,0,8,83,82,0,0,0,0,8,0,2,100],[63,100,104,58,0,0,0,0,13,67,56,87,0,0,0,0,0,55,36,102,0,0,0,0,65,63,100,60,0,0,0,0,0,0,0,0,31,30,93,31,0,0,0,0,100,2,60,84,0,0,0,0,0,106,111,13,0,0,0,0,106,0,83,82],[101,100,90,20,0,0,0,0,13,105,83,87,0,0,0,0,0,35,111,102,0,0,0,0,103,112,100,22,0,0,0,0,0,0,0,0,30,31,73,43,0,0,0,0,2,100,21,17,0,0,0,0,106,0,83,82,0,0,0,0,0,106,111,13] >;
D28.38D4 in GAP, Magma, Sage, TeX
D_{28}._{38}D_4 % in TeX
G:=Group("D28.38D4"); // GroupNames label
G:=SmallGroup(448,735);
// by ID
G=gap.SmallGroup(448,735);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,1123,297,136,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=1,c^4=d^2=a^14,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations