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## G = D28.15D4order 448 = 26·7

### 15th non-split extension by D28 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D28.15D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — Q8.10D14 — D28.15D4
 Lower central C7 — C14 — C2×C28 — D28.15D4
 Upper central C1 — C2 — C2×C4 — C4⋊Q8

Generators and relations for D28.15D4
G = < a,b,c,d | a28=b2=d2=1, c4=a14, bab=cac-1=a-1, dad=a13, cbc-1=a19b, dbd=a26b, dcd=c3 >

Subgroups: 716 in 142 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C42, C4⋊C4, M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C7⋊C8, Dic14, Dic14, C4×D7, D28, D28, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×Q8, D4.10D4, C4.Dic7, Q8⋊D7, C7⋊Q16, C4×C28, C7×C4⋊C4, C4○D28, C4○D28, Q8×D7, Q82D7, Q8×C14, Dic14⋊C4, C28.10D4, C28.C23, C7×C4⋊Q8, Q8.10D14, D28.15D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, D4.10D4, D4×D7, C2×C7⋊D4, C23⋊D14, D28.15D4

Smallest permutation representation of D28.15D4
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)
(1 103)(2 102)(3 101)(4 100)(5 99)(6 98)(7 97)(8 96)(9 95)(10 94)(11 93)(12 92)(13 91)(14 90)(15 89)(16 88)(17 87)(18 86)(19 85)(20 112)(21 111)(22 110)(23 109)(24 108)(25 107)(26 106)(27 105)(28 104)(29 70)(30 69)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 84)(44 83)(45 82)(46 81)(47 80)(48 79)(49 78)(50 77)(51 76)(52 75)(53 74)(54 73)(55 72)(56 71)
(1 104 8 97 15 90 22 111)(2 103 9 96 16 89 23 110)(3 102 10 95 17 88 24 109)(4 101 11 94 18 87 25 108)(5 100 12 93 19 86 26 107)(6 99 13 92 20 85 27 106)(7 98 14 91 21 112 28 105)(29 71 36 64 43 57 50 78)(30 70 37 63 44 84 51 77)(31 69 38 62 45 83 52 76)(32 68 39 61 46 82 53 75)(33 67 40 60 47 81 54 74)(34 66 41 59 48 80 55 73)(35 65 42 58 49 79 56 72)
(1 66)(2 79)(3 64)(4 77)(5 62)(6 75)(7 60)(8 73)(9 58)(10 71)(11 84)(12 69)(13 82)(14 67)(15 80)(16 65)(17 78)(18 63)(19 76)(20 61)(21 74)(22 59)(23 72)(24 57)(25 70)(26 83)(27 68)(28 81)(29 109)(30 94)(31 107)(32 92)(33 105)(34 90)(35 103)(36 88)(37 101)(38 86)(39 99)(40 112)(41 97)(42 110)(43 95)(44 108)(45 93)(46 106)(47 91)(48 104)(49 89)(50 102)(51 87)(52 100)(53 85)(54 98)(55 111)(56 96)```

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

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

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

58 conjugacy classes

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

58 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 C7⋊D4 D4.10D4 D4×D7 D28.15D4 kernel D28.15D4 Dic14⋊C4 C28.10D4 C28.C23 C7×C4⋊Q8 Q8.10D14 Dic14 D28 C2×C28 C4⋊Q8 C42 C2×Q8 C2×C4 C7 C4 C1 # reps 1 2 1 2 1 1 2 2 2 3 3 6 12 2 6 12

Matrix representation of D28.15D4 in GL4(𝔽113) generated by

 64 20 21 77 65 49 30 83 0 0 22 105 0 0 60 91
,
 70 95 6 71 0 0 31 55 69 67 84 29 70 28 41 72
,
 19 38 67 90 0 0 50 96 54 7 51 62 1 14 70 43
,
 108 109 24 91 0 0 2 52 76 83 9 104 84 112 4 109
`G:=sub<GL(4,GF(113))| [64,65,0,0,20,49,0,0,21,30,22,60,77,83,105,91],[70,0,69,70,95,0,67,28,6,31,84,41,71,55,29,72],[19,0,54,1,38,0,7,14,67,50,51,70,90,96,62,43],[108,0,76,84,109,0,83,112,24,2,9,4,91,52,104,109] >;`

D28.15D4 in GAP, Magma, Sage, TeX

`D_{28}._{15}D_4`
`% in TeX`

`G:=Group("D28.15D4");`
`// GroupNames label`

`G:=SmallGroup(448,629);`
`// by ID`

`G=gap.SmallGroup(448,629);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,184,1123,570,297,136,1684,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^28=b^2=d^2=1,c^4=a^14,b*a*b=c*a*c^-1=a^-1,d*a*d=a^13,c*b*c^-1=a^19*b,d*b*d=a^26*b,d*c*d=c^3>;`
`// generators/relations`

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