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## G = C28.3D8order 448 = 26·7

### 3rd non-split extension by C28 of D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — C28.3D8
 Chief series C1 — C7 — C14 — C28 — C56 — C2×C56 — C2×D56 — C28.3D8
 Lower central C7 — C14 — C28 — C56 — C28.3D8
 Upper central C1 — C2 — C2×C4 — C2×C8 — M5(2)

Generators and relations for C28.3D8
G = < a,b,c | a28=1, b8=a14, c2=a7, bab-1=a15, cac-1=a13, cbc-1=a21b7 >

Subgroups: 564 in 62 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C8, C2×C4, D4, C23, D7, C14, C14, C16, C2×C8, M4(2), D8, C2×D4, C28, D14, C2×C14, C8.C4, M5(2), C2×D8, C7⋊C8, C56, D28, C2×C28, C22×D7, M5(2)⋊C2, C112, D56, D56, C4.Dic7, C2×C56, C2×D28, C56.C4, C7×M5(2), C2×D56, C28.3D8
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, D14, D4⋊C4, C4×D7, D28, C7⋊D4, M5(2)⋊C2, C56⋊C2, D56, D14⋊C4, C2.D56, C28.3D8

Smallest permutation representation of C28.3D8
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)
(1 111 48 69 8 104 55 62 15 97 34 83 22 90 41 76)(2 98 49 84 9 91 56 77 16 112 35 70 23 105 42 63)(3 85 50 71 10 106 29 64 17 99 36 57 24 92 43 78)(4 100 51 58 11 93 30 79 18 86 37 72 25 107 44 65)(5 87 52 73 12 108 31 66 19 101 38 59 26 94 45 80)(6 102 53 60 13 95 32 81 20 88 39 74 27 109 46 67)(7 89 54 75 14 110 33 68 21 103 40 61 28 96 47 82)
(1 97 8 104 15 111 22 90)(2 110 9 89 16 96 23 103)(3 95 10 102 17 109 24 88)(4 108 11 87 18 94 25 101)(5 93 12 100 19 107 26 86)(6 106 13 85 20 92 27 99)(7 91 14 98 21 105 28 112)(29 67 36 74 43 81 50 60)(30 80 37 59 44 66 51 73)(31 65 38 72 45 79 52 58)(32 78 39 57 46 64 53 71)(33 63 40 70 47 77 54 84)(34 76 41 83 48 62 55 69)(35 61 42 68 49 75 56 82)```

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

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

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

76 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 7A 7B 7C 8A 8B 8C 8D 8E 14A 14B 14C 14D 14E 14F 16A 16B 16C 16D 28A ··· 28F 28G 28H 28I 56A ··· 56L 56M ··· 56R 112A ··· 112X order 1 2 2 2 2 4 4 7 7 7 8 8 8 8 8 14 14 14 14 14 14 16 16 16 16 28 ··· 28 28 28 28 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 2 56 56 2 2 2 2 2 2 2 4 56 56 2 2 2 4 4 4 4 4 4 4 2 ··· 2 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

76 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C4 D4 D4 D7 D8 SD16 D14 C4×D7 C7⋊D4 D28 D56 C56⋊C2 M5(2)⋊C2 C28.3D8 kernel C28.3D8 C56.C4 C7×M5(2) C2×D56 D56 C56 C2×C28 M5(2) C28 C2×C14 C2×C8 C8 C8 C2×C4 C4 C22 C7 C1 # reps 1 1 1 1 4 1 1 3 2 2 3 6 6 6 12 12 2 12

Matrix representation of C28.3D8 in GL4(𝔽113) generated by

 13 32 0 0 1 46 0 0 32 58 58 81 7 51 32 109
,
 51 33 0 43 47 88 70 55 14 26 104 80 0 42 33 96
,
 107 72 74 1 76 86 72 5 46 54 17 108 24 37 104 16
`G:=sub<GL(4,GF(113))| [13,1,32,7,32,46,58,51,0,0,58,32,0,0,81,109],[51,47,14,0,33,88,26,42,0,70,104,33,43,55,80,96],[107,76,46,24,72,86,54,37,74,72,17,104,1,5,108,16] >;`

C28.3D8 in GAP, Magma, Sage, TeX

`C_{28}._3D_8`
`% in TeX`

`G:=Group("C28.3D8");`
`// GroupNames label`

`G:=SmallGroup(448,73);`
`// by ID`

`G=gap.SmallGroup(448,73);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,85,92,422,387,268,570,1684,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=1,b^8=a^14,c^2=a^7,b*a*b^-1=a^15,c*a*c^-1=a^13,c*b*c^-1=a^21*b^7>;`
`// generators/relations`

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