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## G = D4.9D28order 448 = 26·7

### 4th non-split extension by D4 of D28 acting via D28/D14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D4.9D28
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — D4.10D14 — D4.9D28
 Lower central C7 — C14 — C2×C28 — D4.9D28
 Upper central C1 — C2 — C2×C4 — C4≀C2

Generators and relations for D4.9D28
G = < a,b,c,d | a4=b2=1, c28=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a-1b, dbd-1=ab, dcd-1=c27 >

Subgroups: 764 in 142 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D7, C14, C14, C42, C4⋊C4, M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.10D4, C4≀C2, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, D4.10D4, C56⋊C2, Dic28, C4.Dic7, C4⋊Dic7, D4.D7, C7⋊Q16, C4×C28, C7×M4(2), C2×Dic14, C2×Dic14, C4○D28, C4○D28, D42D7, Q8×D7, C7×C4○D4, Dic14⋊C4, C4.12D28, C7×C4≀C2, C282Q8, C8.D14, D4.9D14, D4.10D14, D4.9D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, D28, C22×D7, D4.10D4, C2×D28, D4×D7, C22⋊D28, D4.9D28

Smallest permutation representation of D4.9D28
On 112 points
Generators in S112
```(1 15 29 43)(2 44 30 16)(3 17 31 45)(4 46 32 18)(5 19 33 47)(6 48 34 20)(7 21 35 49)(8 50 36 22)(9 23 37 51)(10 52 38 24)(11 25 39 53)(12 54 40 26)(13 27 41 55)(14 56 42 28)(57 99 85 71)(58 72 86 100)(59 101 87 73)(60 74 88 102)(61 103 89 75)(62 76 90 104)(63 105 91 77)(64 78 92 106)(65 107 93 79)(66 80 94 108)(67 109 95 81)(68 82 96 110)(69 111 97 83)(70 84 98 112)
(1 36)(2 23)(3 38)(4 25)(5 40)(6 27)(7 42)(8 29)(9 44)(10 31)(11 46)(12 33)(13 48)(14 35)(15 50)(16 37)(17 52)(18 39)(19 54)(20 41)(21 56)(22 43)(24 45)(26 47)(28 49)(30 51)(32 53)(34 55)(57 78)(58 93)(59 80)(60 95)(61 82)(62 97)(63 84)(64 99)(65 86)(66 101)(67 88)(68 103)(69 90)(70 105)(71 92)(72 107)(73 94)(74 109)(75 96)(76 111)(77 98)(79 100)(81 102)(83 104)(85 106)(87 108)(89 110)(91 112)
(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 60 29 88)(2 87 30 59)(3 58 31 86)(4 85 32 57)(5 112 33 84)(6 83 34 111)(7 110 35 82)(8 81 36 109)(9 108 37 80)(10 79 38 107)(11 106 39 78)(12 77 40 105)(13 104 41 76)(14 75 42 103)(15 102 43 74)(16 73 44 101)(17 100 45 72)(18 71 46 99)(19 98 47 70)(20 69 48 97)(21 96 49 68)(22 67 50 95)(23 94 51 66)(24 65 52 93)(25 92 53 64)(26 63 54 91)(27 90 55 62)(28 61 56 89)```

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

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

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

58 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 7A 7B 7C 8A 8B 14A 14B 14C 14D 14E 14F 14G 14H 14I 28A ··· 28F 28G ··· 28U 28V 28W 28X 56A ··· 56F order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 7 7 7 8 8 14 14 14 14 14 14 14 14 14 28 ··· 28 28 ··· 28 28 28 28 56 ··· 56 size 1 1 2 4 28 2 2 4 4 4 28 28 28 56 2 2 2 8 56 2 2 2 4 4 4 8 8 8 2 ··· 2 4 ··· 4 8 8 8 8 ··· 8

58 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D4 D7 D14 D14 D14 D28 D28 D4.10D4 D4×D7 D4×D7 D4.9D28 kernel D4.9D28 Dic14⋊C4 C4.12D28 C7×C4≀C2 C28⋊2Q8 C8.D14 D4.9D14 D4.10D14 Dic14 D28 C2×Dic7 C7×D4 C7×Q8 C4≀C2 C42 M4(2) C4○D4 D4 Q8 C7 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 2 1 1 3 3 3 3 6 6 2 3 3 12

Matrix representation of D4.9D28 in GL4(𝔽113) generated by

 9 46 0 0 67 104 0 0 0 0 104 67 0 0 46 9
,
 0 0 112 0 0 0 0 112 112 0 0 0 0 112 0 0
,
 0 0 112 34 0 0 79 25 104 13 0 0 100 94 0 0
,
 10 3 0 0 4 103 0 0 0 0 48 19 0 0 69 65
`G:=sub<GL(4,GF(113))| [9,67,0,0,46,104,0,0,0,0,104,46,0,0,67,9],[0,0,112,0,0,0,0,112,112,0,0,0,0,112,0,0],[0,0,104,100,0,0,13,94,112,79,0,0,34,25,0,0],[10,4,0,0,3,103,0,0,0,0,48,69,0,0,19,65] >;`

D4.9D28 in GAP, Magma, Sage, TeX

`D_4._9D_{28}`
`% in TeX`

`G:=Group("D4.9D28");`
`// GroupNames label`

`G:=SmallGroup(448,360);`
`// by ID`

`G=gap.SmallGroup(448,360);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,58,1123,136,851,438,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^28=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^-1*b,d*b*d^-1=a*b,d*c*d^-1=c^27>;`
`// generators/relations`

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