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## G = C4.D28order 224 = 25·7

### 5th non-split extension by C4 of D28 acting via D28/C28=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C4.D28
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — D14⋊C4 — C4.D28
 Lower central C7 — C2×C14 — C4.D28
 Upper central C1 — C22 — C42

Generators and relations for C4.D28
G = < a,b,c | a4=b28=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >

Subgroups: 374 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C4.4D4, Dic14, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, D14⋊C4, C4×C28, C2×Dic14, C2×D28, C4.D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, D28, C22×D7, C2×D28, C4○D28, C4.D28

Smallest permutation representation of C4.D28
On 112 points
Generators in S112
```(1 34 102 81)(2 35 103 82)(3 36 104 83)(4 37 105 84)(5 38 106 57)(6 39 107 58)(7 40 108 59)(8 41 109 60)(9 42 110 61)(10 43 111 62)(11 44 112 63)(12 45 85 64)(13 46 86 65)(14 47 87 66)(15 48 88 67)(16 49 89 68)(17 50 90 69)(18 51 91 70)(19 52 92 71)(20 53 93 72)(21 54 94 73)(22 55 95 74)(23 56 96 75)(24 29 97 76)(25 30 98 77)(26 31 99 78)(27 32 100 79)(28 33 101 80)
(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)
(1 101 102 28)(2 27 103 100)(3 99 104 26)(4 25 105 98)(5 97 106 24)(6 23 107 96)(7 95 108 22)(8 21 109 94)(9 93 110 20)(10 19 111 92)(11 91 112 18)(12 17 85 90)(13 89 86 16)(14 15 87 88)(29 57 76 38)(30 37 77 84)(31 83 78 36)(32 35 79 82)(33 81 80 34)(39 75 58 56)(40 55 59 74)(41 73 60 54)(42 53 61 72)(43 71 62 52)(44 51 63 70)(45 69 64 50)(46 49 65 68)(47 67 66 48)```

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 D4 D7 C4○D4 D14 D28 C4○D28 kernel C4.D28 D14⋊C4 C4×C28 C2×Dic14 C2×D28 C28 C42 C14 C2×C4 C4 C2 # reps 1 4 1 1 1 2 3 4 9 12 24

Matrix representation of C4.D28 in GL4(𝔽29) generated by

 7 5 0 0 19 22 0 0 0 0 18 11 0 0 10 11
,
 28 26 0 0 6 19 0 0 0 0 12 0 0 0 0 12
,
 23 22 0 0 26 6 0 0 0 0 12 0 0 0 24 17
`G:=sub<GL(4,GF(29))| [7,19,0,0,5,22,0,0,0,0,18,10,0,0,11,11],[28,6,0,0,26,19,0,0,0,0,12,0,0,0,0,12],[23,26,0,0,22,6,0,0,0,0,12,24,0,0,0,17] >;`

C4.D28 in GAP, Magma, Sage, TeX

`C_4.D_{28}`
`% in TeX`

`G:=Group("C4.D28");`
`// GroupNames label`

`G:=SmallGroup(224,70);`
`// by ID`

`G=gap.SmallGroup(224,70);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,55,218,86,6917]);`
`// Polycyclic`

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

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