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G = C4⋊D28order 224 = 25·7

The semidirect product of C4 and D28 acting via D28/D14=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4⋊D28
G = < a,b,c | a28=b4=c2=1, bab-1=a15, cac=a-1, cbc=b-1 >

Subgroups: 502 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, D7, C14, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, C28, C28, D14, D14, C2×C14, C4⋊D4, C4×D7, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, D14⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, C2×D28, C4⋊D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, D28, C22×D7, C2×D28, D4×D7, Q82D7, C4⋊D28

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D7 C4○D4 D14 D28 D4×D7 Q8⋊2D7 kernel C4⋊D28 D14⋊C4 C7×C4⋊C4 C2×C4×D7 C2×D28 C28 D14 C4⋊C4 C14 C2×C4 C4 C2 C2 # reps 1 2 1 1 3 2 2 3 2 9 12 3 3

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

 19 7 0 0 22 28 0 0 0 0 21 13 0 0 24 8
,
 21 6 0 0 23 8 0 0 0 0 28 0 0 0 1 1
,
 19 7 0 0 19 10 0 0 0 0 1 0 0 0 28 28
`G:=sub<GL(4,GF(29))| [19,22,0,0,7,28,0,0,0,0,21,24,0,0,13,8],[21,23,0,0,6,8,0,0,0,0,28,1,0,0,0,1],[19,19,0,0,7,10,0,0,0,0,1,28,0,0,0,28] >;`

C4⋊D28 in GAP, Magma, Sage, TeX

`C_4\rtimes D_{28}`
`% in TeX`

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

`G:=SmallGroup(224,90);`
`// by ID`

`G=gap.SmallGroup(224,90);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,188,50,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=b^4=c^2=1,b*a*b^-1=a^15,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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