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## G = D28.3D4order 448 = 26·7

### 3rd non-split extension by D28 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D28.3D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — D28.C4 — D28.3D4
 Lower central C7 — C14 — C2×C28 — D28.3D4
 Upper central C1 — C2 — C2×C4 — C4.D4

Generators and relations for D28.3D4
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=a21c3 >

Subgroups: 652 in 108 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, C2×D4, C2×D4, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C4.D4, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C7⋊C8, C7⋊C8, C56, Dic14, C4×D7, D28, D28, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, D4.4D4, C8×D7, C8⋊D7, C56⋊C2, D56, C2×C7⋊C8, C4.Dic7, D4⋊D7, D4.D7, C7×M4(2), C2×D28, C4○D28, D4×C14, C28.53D4, C28.46D4, C7×C4.D4, D28.C4, C8⋊D14, C2×D4⋊D7, D4.D14, D28.3D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C22×D7, D4.4D4, C4○D28, D4×D7, D14⋊D4, D28.3D4

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

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

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

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

49 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 14A 14B 14C 14D 14E 14F 14G ··· 14L 28A ··· 28F 56A ··· 56L order 1 2 2 2 2 2 4 4 4 7 7 7 8 8 8 8 8 8 8 14 14 14 14 14 14 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 2 8 28 56 2 2 28 2 2 2 4 4 8 14 14 28 56 2 2 2 4 4 4 8 ··· 8 4 ··· 4 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 D4.4D4 D4×D7 D28.3D4 kernel D28.3D4 C28.53D4 C28.46D4 C7×C4.D4 D28.C4 C8⋊D14 C2×D4⋊D7 D4.D14 C7⋊C8 Dic14 D28 C4.D4 C2×C14 M4(2) C2×D4 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 D28.3D4 in GL8(𝔽113)

 33 0 34 0 0 0 0 0 0 33 0 34 0 0 0 0 71 0 104 0 0 0 0 0 0 71 0 104 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 91 112 0 0 0 0 0 0 62 0 112 2 0 0 0 0 0 28 112 1
,
 80 0 1 0 0 0 0 0 0 80 0 1 0 0 0 0 42 0 33 0 0 0 0 0 0 42 0 33 0 0 0 0 0 0 0 0 112 0 62 51 0 0 0 0 0 0 109 0 0 0 0 0 0 28 0 0 0 0 0 0 0 28 112 1
,
 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 109 0 0 91 0 0 0 0 1 72 0 0 0 0 0 0 1 36 82 0
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 112 0 0 51 0 0 0 0 0 0 4 0 0 0 0 0 62 85 0 0 0 0 0 0 62 85 0 1

`G:=sub<GL(8,GF(113))| [33,0,71,0,0,0,0,0,0,33,0,71,0,0,0,0,34,0,104,0,0,0,0,0,0,34,0,104,0,0,0,0,0,0,0,0,1,91,62,0,0,0,0,0,72,112,0,28,0,0,0,0,0,0,112,112,0,0,0,0,0,0,2,1],[80,0,42,0,0,0,0,0,0,80,0,42,0,0,0,0,1,0,33,0,0,0,0,0,0,1,0,33,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,28,28,0,0,0,0,62,109,0,112,0,0,0,0,51,0,0,1],[0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,109,1,1,0,0,0,0,0,0,72,36,0,0,0,0,1,0,0,82,0,0,0,0,0,91,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,112,0,62,62,0,0,0,0,0,0,85,85,0,0,0,0,0,4,0,0,0,0,0,0,51,0,0,1] >;`

D28.3D4 in GAP, Magma, Sage, TeX

`D_{28}._3D_4`
`% in TeX`

`G:=Group("D28.3D4");`
`// GroupNames label`

`G:=SmallGroup(448,283);`
`// by ID`

`G=gap.SmallGroup(448,283);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,64,590,555,297,136,1684,851,18822]);`
`// 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^21*c^3>;`
`// generators/relations`

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