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G = D28.5D4order 448 = 26·7

5th non-split extension by D28 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D28.5D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — Q8.10D14 — D28.5D4
 Lower central C7 — C14 — C2×C28 — D28.5D4
 Upper central C1 — C2 — C2×C4 — C4.10D4

Generators and relations for D28.5D4
G = < a,b,c,d | a28=b2=1, c4=a14, d2=a21, bab=a-1, cac-1=a15, ad=da, cbc-1=a7b, dbd-1=a21b, dcd-1=a21c3 >

Subgroups: 940 in 146 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×Q8, C22×D7, D4.8D4, C56⋊C2, D56, C4×Dic7, D14⋊C4, C7×M4(2), C2×D28, C4○D28, C4○D28, Q8×D7, Q82D7, Q8×C14, D284C4, C7×C4.10D4, C8⋊D14, C28.23D4, Q8.10D14, D28.5D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, D28, C22×D7, D4.8D4, C2×D28, D4×D7, C22⋊D28, D28.5D4

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

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

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

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

49 conjugacy classes

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

49 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 8 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 D28 D4.8D4 D4×D7 D28.5D4 kernel D28.5D4 D28⋊4C4 C7×C4.10D4 C8⋊D14 C28.23D4 Q8.10D14 Dic14 D28 C2×C28 C4.10D4 M4(2) C2×Q8 C2×C4 C7 C4 C1 # reps 1 2 1 2 1 1 2 2 2 3 6 3 12 2 6 3

Matrix representation of D28.5D4 in GL8(𝔽113)

 34 0 89 0 0 0 0 0 0 34 0 89 0 0 0 0 59 0 88 0 0 0 0 0 0 59 0 88 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 98 0 0 0 0 0 0 0 0 98 0 0 0 0 0 0 0 0 15
,
 0 0 10 0 0 0 0 0 0 0 0 10 0 0 0 0 34 0 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 98 0 0 0 0 0 0 0 0 0 0 98 0 0 0 0 0 0 15 0
,
 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 15 0
,
 112 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 15 0 0 0 0 0 0 112 0 0 0 0 0 0 98 0 0 0

`G:=sub<GL(8,GF(113))| [34,0,59,0,0,0,0,0,0,34,0,59,0,0,0,0,89,0,88,0,0,0,0,0,0,89,0,88,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,15],[0,0,34,0,0,0,0,0,0,0,0,34,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,98,0],[0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,112,0],[112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,112,0,0,0,0,0,0,15,0,0,0,0,0,0,112,0,0,0] >;`

D28.5D4 in GAP, Magma, Sage, TeX

`D_{28}._5D_4`
`% in TeX`

`G:=Group("D28.5D4");`
`// GroupNames label`

`G:=SmallGroup(448,287);`
`// by ID`

`G=gap.SmallGroup(448,287);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,226,1123,570,136,1684,438,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^28=b^2=1,c^4=a^14,d^2=a^21,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^21*c^3>;`
`// generators/relations`

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