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## G = D28⋊C4order 224 = 25·7

### 5th semidirect product of D28 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for D28⋊C4
G = < a,b,c | a28=b2=c4=1, bab=a-1, cac-1=a15, cbc-1=a14b >

Subgroups: 406 in 94 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C4×D4, C4×D7, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C4×Dic7, D14⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, D28⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C4×D7, C22×D7, C2×C4×D7, D4×D7, Q82D7, D28⋊C4

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

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4F 4G 4H 4I 4J 4K 4L 7A 7B 7C 14A ··· 14I 28A ··· 28R order 1 2 2 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 14 14 14 14 2 ··· 2 7 7 7 7 14 14 2 2 2 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D7 C4○D4 D14 C4×D7 D4×D7 Q8⋊2D7 kernel D28⋊C4 C4×Dic7 D14⋊C4 C7×C4⋊C4 C2×C4×D7 C2×D28 D28 Dic7 C4⋊C4 C14 C2×C4 C4 C2 C2 # reps 1 1 2 1 2 1 8 2 3 2 9 12 3 3

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

 10 7 0 0 22 1 0 0 0 0 24 11 0 0 24 5
,
 19 22 0 0 10 10 0 0 0 0 28 0 0 0 7 1
,
 12 0 0 0 0 12 0 0 0 0 27 16 0 0 16 2
`G:=sub<GL(4,GF(29))| [10,22,0,0,7,1,0,0,0,0,24,24,0,0,11,5],[19,10,0,0,22,10,0,0,0,0,28,7,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,27,16,0,0,16,2] >;`

D28⋊C4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(224,88);`
`// by ID`

`G=gap.SmallGroup(224,88);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,103,188,50,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^15,c*b*c^-1=a^14*b>;`
`// generators/relations`

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