Copied to
clipboard

## G = D4.6D28order 448 = 26·7

### 1st non-split extension by D4 of D28 acting via D28/D14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D4.6D28
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C4×D7 — C2×D4×D7 — D4.6D28
 Lower central C7 — C14 — C2×C28 — D4.6D28
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for D4.6D28
G = < a,b,c,d | a28=b2=c4=1, d2=a7, bab=a-1, cac-1=a15, ad=da, cbc-1=a7b, dbd-1=a21b, dcd-1=a7c-1 >

Subgroups: 1332 in 188 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, C7⋊C8, C56, Dic14, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C22×D7, C22×D7, C22×C14, C22⋊SD16, C56⋊C2, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, D4.D7, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, D4×D7, C2×C7⋊D4, D4×C14, C23×D7, C14.D8, D14⋊C8, C7×D4⋊C4, D142Q8, C2×C56⋊C2, C2×D4.D7, C2×D4×D7, D4.6D28
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C22≀C2, C2×SD16, C8⋊C22, D28, C22×D7, C22⋊SD16, C2×D28, D4×D7, C22⋊D28, D8⋊D7, D7×SD16, D4.6D28

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

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

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

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

61 conjugacy classes

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

61 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D7 SD16 D14 D14 D14 D28 C8⋊C22 D4×D7 D4×D7 D8⋊D7 D7×SD16 kernel D4.6D28 C14.D8 D14⋊C8 C7×D4⋊C4 D14⋊2Q8 C2×C56⋊C2 C2×D4.D7 C2×D4×D7 D28 C2×Dic7 C7×D4 C22×D7 D4⋊C4 D14 C4⋊C4 C2×C8 C2×D4 D4 C14 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 1 2 1 3 4 3 3 3 12 1 3 3 6 6

Matrix representation of D4.6D28 in GL4(𝔽113) generated by

 112 72 0 0 91 1 0 0 0 0 89 79 0 0 10 0
,
 1 0 0 0 22 112 0 0 0 0 0 79 0 0 103 0
,
 26 81 0 0 60 87 0 0 0 0 17 67 0 0 80 96
,
 0 81 0 0 60 87 0 0 0 0 96 46 0 0 33 17
`G:=sub<GL(4,GF(113))| [112,91,0,0,72,1,0,0,0,0,89,10,0,0,79,0],[1,22,0,0,0,112,0,0,0,0,0,103,0,0,79,0],[26,60,0,0,81,87,0,0,0,0,17,80,0,0,67,96],[0,60,0,0,81,87,0,0,0,0,96,33,0,0,46,17] >;`

D4.6D28 in GAP, Magma, Sage, TeX

`D_4._6D_{28}`
`% in TeX`

`G:=Group("D4.6D28");`
`// GroupNames label`

`G:=SmallGroup(448,310);`
`// by ID`

`G=gap.SmallGroup(448,310);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,135,268,570,297,136,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^28=b^2=c^4=1,d^2=a^7,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^7*b,d*b*d^-1=a^21*b,d*c*d^-1=a^7*c^-1>;`
`// generators/relations`

׿
×
𝔽