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## G = C28.48D4order 224 = 25·7

### 5th non-split extension by C28 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C28.48D4
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C2×Dic14 — C28.48D4
 Lower central C7 — C2×C14 — C28.48D4
 Upper central C1 — C22 — C22×C4

Generators and relations for C28.48D4
G = < a,b,c | a28=b4=1, c2=a14, bab-1=cac-1=a-1, cbc-1=a14b-1 >

Subgroups: 238 in 74 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊Q8, Dic14, C2×Dic7, C2×C28, C2×C28, C22×C14, Dic7⋊C4, C4⋊Dic7, C23.D7, C2×Dic14, C22×C28, C28.48D4
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, Dic14, C7⋊D4, C22×D7, C2×Dic14, C4○D28, C2×C7⋊D4, C28.48D4

Smallest permutation representation of C28.48D4
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)
(1 104 33 63)(2 103 34 62)(3 102 35 61)(4 101 36 60)(5 100 37 59)(6 99 38 58)(7 98 39 57)(8 97 40 84)(9 96 41 83)(10 95 42 82)(11 94 43 81)(12 93 44 80)(13 92 45 79)(14 91 46 78)(15 90 47 77)(16 89 48 76)(17 88 49 75)(18 87 50 74)(19 86 51 73)(20 85 52 72)(21 112 53 71)(22 111 54 70)(23 110 55 69)(24 109 56 68)(25 108 29 67)(26 107 30 66)(27 106 31 65)(28 105 32 64)
(1 77 15 63)(2 76 16 62)(3 75 17 61)(4 74 18 60)(5 73 19 59)(6 72 20 58)(7 71 21 57)(8 70 22 84)(9 69 23 83)(10 68 24 82)(11 67 25 81)(12 66 26 80)(13 65 27 79)(14 64 28 78)(29 94 43 108)(30 93 44 107)(31 92 45 106)(32 91 46 105)(33 90 47 104)(34 89 48 103)(35 88 49 102)(36 87 50 101)(37 86 51 100)(38 85 52 99)(39 112 53 98)(40 111 54 97)(41 110 55 96)(42 109 56 95)```

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 14A ··· 14U 28A ··· 28X order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 2 2 2 2 28 28 28 28 2 2 2 2 ··· 2 2 ··· 2

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 D4 Q8 D7 C4○D4 D14 D14 C7⋊D4 Dic14 C4○D28 kernel C28.48D4 Dic7⋊C4 C4⋊Dic7 C23.D7 C2×Dic14 C22×C28 C28 C2×C14 C22×C4 C14 C2×C4 C23 C4 C22 C2 # reps 1 2 1 2 1 1 2 2 3 2 6 3 12 12 12

Matrix representation of C28.48D4 in GL4(𝔽29) generated by

 21 0 0 0 23 18 0 0 0 0 16 0 0 0 0 20
,
 22 11 0 0 22 7 0 0 0 0 0 1 0 0 28 0
,
 22 11 0 0 6 7 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(29))| [21,23,0,0,0,18,0,0,0,0,16,0,0,0,0,20],[22,22,0,0,11,7,0,0,0,0,0,28,0,0,1,0],[22,6,0,0,11,7,0,0,0,0,0,1,0,0,1,0] >;`

C28.48D4 in GAP, Magma, Sage, TeX

`C_{28}._{48}D_4`
`% in TeX`

`G:=Group("C28.48D4");`
`// GroupNames label`

`G:=SmallGroup(224,119);`
`// by ID`

`G=gap.SmallGroup(224,119);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,217,103,218,6917]);`
`// Polycyclic`

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

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