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## G = D28.6D4order 448 = 26·7

### 6th non-split extension by D28 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D28.6D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — D28.C4 — D28.6D4
 Lower central C7 — C14 — C2×C28 — D28.6D4
 Upper central C1 — C2 — C2×C4 — C4.10D4

Generators and relations for D28.6D4
G = < a,b,c,d | a28=b2=1, c4=a14, d2=a7, bab=a-1, cac-1=a15, ad=da, cbc-1=a14b, dbd-1=a7b, dcd-1=a7c3 >

Subgroups: 620 in 104 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C7⋊C8, C7⋊C8, C56, Dic14, C4×D7, D28, D28, C7⋊D4, C2×C28, C2×C28, C7×Q8, C22×D7, D4.3D4, C8×D7, C8⋊D7, C56⋊C2, D56, C2×C7⋊C8, C4.Dic7, Q8⋊D7, C7⋊Q16, C7×M4(2), C2×D28, C4○D28, Q8×C14, C28.53D4, C28.46D4, C7×C4.10D4, D28.C4, C8⋊D14, C2×Q8⋊D7, C28.C23, D28.6D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C22×D7, D4.3D4, C4○D28, D4×D7, D14⋊D4, D28.6D4

Smallest permutation representation of D28.6D4
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 96)(2 95)(3 94)(4 93)(5 92)(6 91)(7 90)(8 89)(9 88)(10 87)(11 86)(12 85)(13 112)(14 111)(15 110)(16 109)(17 108)(18 107)(19 106)(20 105)(21 104)(22 103)(23 102)(24 101)(25 100)(26 99)(27 98)(28 97)(29 83)(30 82)(31 81)(32 80)(33 79)(34 78)(35 77)(36 76)(37 75)(38 74)(39 73)(40 72)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(56 84)
(1 63 22 70 15 77 8 84)(2 78 23 57 16 64 9 71)(3 65 24 72 17 79 10 58)(4 80 25 59 18 66 11 73)(5 67 26 74 19 81 12 60)(6 82 27 61 20 68 13 75)(7 69 28 76 21 83 14 62)(29 97 50 104 43 111 36 90)(30 112 51 91 44 98 37 105)(31 99 52 106 45 85 38 92)(32 86 53 93 46 100 39 107)(33 101 54 108 47 87 40 94)(34 88 55 95 48 102 41 109)(35 103 56 110 49 89 42 96)
(1 104 8 111 15 90 22 97)(2 105 9 112 16 91 23 98)(3 106 10 85 17 92 24 99)(4 107 11 86 18 93 25 100)(5 108 12 87 19 94 26 101)(6 109 13 88 20 95 27 102)(7 110 14 89 21 96 28 103)(29 84 36 63 43 70 50 77)(30 57 37 64 44 71 51 78)(31 58 38 65 45 72 52 79)(32 59 39 66 46 73 53 80)(33 60 40 67 47 74 54 81)(34 61 41 68 48 75 55 82)(35 62 42 69 49 76 56 83)```

`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), (1,96)(2,95)(3,94)(4,93)(5,92)(6,91)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,97)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,84), (1,63,22,70,15,77,8,84)(2,78,23,57,16,64,9,71)(3,65,24,72,17,79,10,58)(4,80,25,59,18,66,11,73)(5,67,26,74,19,81,12,60)(6,82,27,61,20,68,13,75)(7,69,28,76,21,83,14,62)(29,97,50,104,43,111,36,90)(30,112,51,91,44,98,37,105)(31,99,52,106,45,85,38,92)(32,86,53,93,46,100,39,107)(33,101,54,108,47,87,40,94)(34,88,55,95,48,102,41,109)(35,103,56,110,49,89,42,96), (1,104,8,111,15,90,22,97)(2,105,9,112,16,91,23,98)(3,106,10,85,17,92,24,99)(4,107,11,86,18,93,25,100)(5,108,12,87,19,94,26,101)(6,109,13,88,20,95,27,102)(7,110,14,89,21,96,28,103)(29,84,36,63,43,70,50,77)(30,57,37,64,44,71,51,78)(31,58,38,65,45,72,52,79)(32,59,39,66,46,73,53,80)(33,60,40,67,47,74,54,81)(34,61,41,68,48,75,55,82)(35,62,42,69,49,76,56,83)>;`

`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), (1,96)(2,95)(3,94)(4,93)(5,92)(6,91)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,97)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,84), (1,63,22,70,15,77,8,84)(2,78,23,57,16,64,9,71)(3,65,24,72,17,79,10,58)(4,80,25,59,18,66,11,73)(5,67,26,74,19,81,12,60)(6,82,27,61,20,68,13,75)(7,69,28,76,21,83,14,62)(29,97,50,104,43,111,36,90)(30,112,51,91,44,98,37,105)(31,99,52,106,45,85,38,92)(32,86,53,93,46,100,39,107)(33,101,54,108,47,87,40,94)(34,88,55,95,48,102,41,109)(35,103,56,110,49,89,42,96), (1,104,8,111,15,90,22,97)(2,105,9,112,16,91,23,98)(3,106,10,85,17,92,24,99)(4,107,11,86,18,93,25,100)(5,108,12,87,19,94,26,101)(6,109,13,88,20,95,27,102)(7,110,14,89,21,96,28,103)(29,84,36,63,43,70,50,77)(30,57,37,64,44,71,51,78)(31,58,38,65,45,72,52,79)(32,59,39,66,46,73,53,80)(33,60,40,67,47,74,54,81)(34,61,41,68,48,75,55,82)(35,62,42,69,49,76,56,83) );`

`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)], [(1,96),(2,95),(3,94),(4,93),(5,92),(6,91),(7,90),(8,89),(9,88),(10,87),(11,86),(12,85),(13,112),(14,111),(15,110),(16,109),(17,108),(18,107),(19,106),(20,105),(21,104),(22,103),(23,102),(24,101),(25,100),(26,99),(27,98),(28,97),(29,83),(30,82),(31,81),(32,80),(33,79),(34,78),(35,77),(36,76),(37,75),(38,74),(39,73),(40,72),(41,71),(42,70),(43,69),(44,68),(45,67),(46,66),(47,65),(48,64),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(56,84)], [(1,63,22,70,15,77,8,84),(2,78,23,57,16,64,9,71),(3,65,24,72,17,79,10,58),(4,80,25,59,18,66,11,73),(5,67,26,74,19,81,12,60),(6,82,27,61,20,68,13,75),(7,69,28,76,21,83,14,62),(29,97,50,104,43,111,36,90),(30,112,51,91,44,98,37,105),(31,99,52,106,45,85,38,92),(32,86,53,93,46,100,39,107),(33,101,54,108,47,87,40,94),(34,88,55,95,48,102,41,109),(35,103,56,110,49,89,42,96)], [(1,104,8,111,15,90,22,97),(2,105,9,112,16,91,23,98),(3,106,10,85,17,92,24,99),(4,107,11,86,18,93,25,100),(5,108,12,87,19,94,26,101),(6,109,13,88,20,95,27,102),(7,110,14,89,21,96,28,103),(29,84,36,63,43,70,50,77),(30,57,37,64,44,71,51,78),(31,58,38,65,45,72,52,79),(32,59,39,66,46,73,53,80),(33,60,40,67,47,74,54,81),(34,61,41,68,48,75,55,82),(35,62,42,69,49,76,56,83)]])`

49 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 14A 14B 14C 14D 14E 14F 28A ··· 28F 28G ··· 28L 56A ··· 56L order 1 2 2 2 2 4 4 4 4 7 7 7 8 8 8 8 8 8 8 14 14 14 14 14 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 28 56 2 2 8 28 2 2 2 4 4 8 14 14 28 56 2 2 2 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8

49 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 8 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D7 C4○D4 D14 D14 C4○D28 D4.3D4 D4×D7 D28.6D4 kernel D28.6D4 C28.53D4 C28.46D4 C7×C4.10D4 D28.C4 C8⋊D14 C2×Q8⋊D7 C28.C23 C7⋊C8 Dic14 D28 C4.10D4 C2×C14 M4(2) C2×Q8 C22 C7 C4 C1 # reps 1 1 1 1 1 1 1 1 2 1 1 3 2 6 3 12 2 6 3

Matrix representation of D28.6D4 in GL6(𝔽113)

 112 89 0 0 0 0 24 10 0 0 0 0 0 0 1 87 0 0 0 0 87 112 0 0 0 0 62 68 112 77 0 0 89 77 44 1
,
 1 0 0 0 0 0 89 112 0 0 0 0 0 0 0 23 0 87 0 0 1 36 69 111 0 0 96 21 44 99 0 0 53 84 48 33
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 69 69 0 0 0 0 23 0 87 0 0 0 105 112 26 0 0 1 3 0 90
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 87 104 14 26 0 0 112 77 44 2 0 0 98 5 95 14 0 0 86 29 65 80

`G:=sub<GL(6,GF(113))| [112,24,0,0,0,0,89,10,0,0,0,0,0,0,1,87,62,89,0,0,87,112,68,77,0,0,0,0,112,44,0,0,0,0,77,1],[1,89,0,0,0,0,0,112,0,0,0,0,0,0,0,1,96,53,0,0,23,36,21,84,0,0,0,69,44,48,0,0,87,111,99,33],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,1,0,0,69,23,105,3,0,0,69,0,112,0,0,0,0,87,26,90],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,87,112,98,86,0,0,104,77,5,29,0,0,14,44,95,65,0,0,26,2,14,80] >;`

D28.6D4 in GAP, Magma, Sage, TeX

`D_{28}._6D_4`
`% in TeX`

`G:=Group("D28.6D4");`
`// GroupNames label`

`G:=SmallGroup(448,288);`
`// by ID`

`G=gap.SmallGroup(448,288);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,64,590,219,184,297,136,1684,851,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^28=b^2=1,c^4=a^14,d^2=a^7,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^14*b,d*b*d^-1=a^7*b,d*c*d^-1=a^7*c^3>;`
`// generators/relations`

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