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G = D28.40D4order 448 = 26·7

10th non-split extension by D28 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D28.40D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — D4.10D14 — D28.40D4
 Lower central C7 — C14 — C2×C28 — D28.40D4
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for D28.40D4
G = < a,b,c,d | a28=b2=1, c4=d2=a14, bab=a-1, cac-1=a15, ad=da, cbc-1=a7b, bd=db, dcd-1=a14c3 >

Subgroups: 716 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, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 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, C7×Q8, D4.10D4, C4.Dic7, C4×Dic7, Dic7⋊C4, Q8⋊D7, C7⋊Q16, C7×M4(2), C7×SD16, C7×Q16, C2×Dic14, C2×Dic14, C4○D28, C4○D28, D42D7, Q8×D7, Q8×C14, C7×C4○D4, C4.12D28, D284C4, D42Dic7, C28.C23, Dic7⋊Q8, C7×C8.C22, D4.10D14, D28.40D4
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, D28.40D4

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

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

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

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

49 conjugacy classes

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

49 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 8 type + + + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D4 D7 D14 D14 D14 C7⋊D4 C7⋊D4 D4.10D4 D4×D7 D4×D7 D28.40D4 kernel D28.40D4 C4.12D28 D28⋊4C4 D4⋊2Dic7 C28.C23 Dic7⋊Q8 C7×C8.C22 D4.10D14 Dic14 D28 C2×Dic7 C7×D4 C7×Q8 C8.C22 M4(2) C2×Q8 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 3

Matrix representation of D28.40D4 in GL6(𝔽113)

 85 97 0 0 0 0 0 4 0 0 0 0 0 0 112 41 0 0 0 0 22 1 0 0 0 0 29 48 1 22 0 0 65 70 41 112
,
 85 97 0 0 0 0 56 28 0 0 0 0 0 0 10 78 52 73 0 0 90 36 40 17 0 0 84 65 112 91 0 0 64 16 101 68
,
 112 112 0 0 0 0 0 1 0 0 0 0 0 0 64 42 6 59 0 0 84 57 54 41 0 0 12 7 112 0 0 0 40 46 49 106
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 106 90 72 0 0 0 0 112 111 91 0 0 15 10 66 84 0 0 50 30 66 55

`G:=sub<GL(6,GF(113))| [85,0,0,0,0,0,97,4,0,0,0,0,0,0,112,22,29,65,0,0,41,1,48,70,0,0,0,0,1,41,0,0,0,0,22,112],[85,56,0,0,0,0,97,28,0,0,0,0,0,0,10,90,84,64,0,0,78,36,65,16,0,0,52,40,112,101,0,0,73,17,91,68],[112,0,0,0,0,0,112,1,0,0,0,0,0,0,64,84,12,40,0,0,42,57,7,46,0,0,6,54,112,49,0,0,59,41,0,106],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,106,0,15,50,0,0,90,112,10,30,0,0,72,111,66,66,0,0,0,91,84,55] >;`

D28.40D4 in GAP, Magma, Sage, TeX

`D_{28}._{40}D_4`
`% in TeX`

`G:=Group("D28.40D4");`
`// GroupNames label`

`G:=SmallGroup(448,739);`
`// by ID`

`G=gap.SmallGroup(448,739);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,184,1123,297,136,851,438,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^28=b^2=1,c^4=d^2=a^14,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d^-1=a^14*c^3>;`
`// generators/relations`

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