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## G = C28.17D4order 224 = 25·7

### 17th non-split extension by C28 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C28.17D4
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C4×Dic7 — C28.17D4
 Lower central C7 — C2×C14 — C28.17D4
 Upper central C1 — C22 — C2×D4

Generators and relations for C28.17D4
G = < a,b,c | a28=b4=1, c2=a14, bab-1=a13, cac-1=a-1, cbc-1=a14b-1 >

Subgroups: 254 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C42, C22⋊C4, C2×D4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C4.4D4, Dic14, C2×Dic7, C2×C28, C7×D4, C22×C14, C4×Dic7, C23.D7, C2×Dic14, D4×C14, C28.17D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, C7⋊D4, C22×D7, D42D7, C2×C7⋊D4, C28.17D4

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 D4 D7 C4○D4 D14 D14 C7⋊D4 D4⋊2D7 kernel C28.17D4 C4×Dic7 C23.D7 C2×Dic14 D4×C14 C28 C2×D4 C14 C2×C4 C23 C4 C2 # reps 1 1 4 1 1 2 3 4 3 6 12 6

Matrix representation of C28.17D4 in GL4(𝔽29) generated by

 24 0 0 0 0 23 0 0 0 0 28 22 0 0 21 1
,
 0 1 0 0 28 0 0 0 0 0 17 3 0 0 20 12
,
 0 28 0 0 28 0 0 0 0 0 12 0 0 0 9 17
`G:=sub<GL(4,GF(29))| [24,0,0,0,0,23,0,0,0,0,28,21,0,0,22,1],[0,28,0,0,1,0,0,0,0,0,17,20,0,0,3,12],[0,28,0,0,28,0,0,0,0,0,12,9,0,0,0,17] >;`

C28.17D4 in GAP, Magma, Sage, TeX

`C_{28}._{17}D_4`
`% in TeX`

`G:=Group("C28.17D4");`
`// GroupNames label`

`G:=SmallGroup(224,131);`
`// by ID`

`G=gap.SmallGroup(224,131);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,55,506,116,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=b^4=1,c^2=a^14,b*a*b^-1=a^13,c*a*c^-1=a^-1,c*b*c^-1=a^14*b^-1>;`
`// generators/relations`

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