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## G = D28.C4order 224 = 25·7

### The non-split extension by D28 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D28.C4
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C4○D28 — D28.C4
 Lower central C7 — C14 — D28.C4
 Upper central C1 — C4 — M4(2)

Generators and relations for D28.C4
G = < a,b,c | a28=b2=1, c4=a14, bab=a-1, cac-1=a15, cbc-1=a14b >

Subgroups: 206 in 62 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C2×C8, M4(2), M4(2), C4○D4, Dic7, C28, D14, C2×C14, C8○D4, C7⋊C8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C8×D7, C8⋊D7, C2×C7⋊C8, C7×M4(2), C4○D28, D28.C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, D14, C8○D4, C4×D7, C22×D7, C2×C4×D7, 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 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(50 56)(51 55)(52 54)(57 59)(60 84)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(85 103)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(104 112)(105 111)(106 110)(107 109)
(1 98 83 43 15 112 69 29)(2 85 84 30 16 99 70 44)(3 100 57 45 17 86 71 31)(4 87 58 32 18 101 72 46)(5 102 59 47 19 88 73 33)(6 89 60 34 20 103 74 48)(7 104 61 49 21 90 75 35)(8 91 62 36 22 105 76 50)(9 106 63 51 23 92 77 37)(10 93 64 38 24 107 78 52)(11 108 65 53 25 94 79 39)(12 95 66 40 26 109 80 54)(13 110 67 55 27 96 81 41)(14 97 68 42 28 111 82 56)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 14A 14B 14C 14D 14E 14F 28A ··· 28F 28G 28H 28I 56A ··· 56L order 1 2 2 2 2 4 4 4 4 4 7 7 7 8 8 8 8 8 8 8 8 8 8 14 14 14 14 14 14 28 ··· 28 28 28 28 56 ··· 56 size 1 1 2 14 14 1 1 2 14 14 2 2 2 2 2 2 2 7 7 7 7 14 14 2 2 2 4 4 4 2 ··· 2 4 4 4 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D7 D14 D14 C8○D4 C4×D7 C4×D7 D28.C4 kernel D28.C4 C8×D7 C8⋊D7 C2×C7⋊C8 C7×M4(2) C4○D28 Dic14 D28 C7⋊D4 M4(2) C8 C2×C4 C7 C4 C22 C1 # reps 1 2 2 1 1 1 2 2 4 3 6 3 4 6 6 6

Matrix representation of D28.C4 in GL4(𝔽113) generated by

 33 42 0 0 79 104 0 0 0 0 0 69 0 0 18 0
,
 0 104 0 0 25 0 0 0 0 0 1 0 0 0 0 112
,
 15 0 0 0 0 15 0 0 0 0 0 112 0 0 15 0
`G:=sub<GL(4,GF(113))| [33,79,0,0,42,104,0,0,0,0,0,18,0,0,69,0],[0,25,0,0,104,0,0,0,0,0,1,0,0,0,0,112],[15,0,0,0,0,15,0,0,0,0,0,15,0,0,112,0] >;`

D28.C4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(224,102);`
`// by ID`

`G=gap.SmallGroup(224,102);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,188,50,69,6917]);`
`// Polycyclic`

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

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