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## G = D28.31D4order 448 = 26·7

### 1st non-split extension by D28 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D28.31D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×D28 — C22×D28 — D28.31D4
 Lower central C7 — C14 — C2×C28 — D28.31D4
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for D28.31D4
G = < a,b,c,d | a28=b2=d2=1, c4=a14, bab=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd=a7c3 >

Subgroups: 1404 in 188 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C24, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, C56, Dic14, D28, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×C14, C22⋊SD16, C56⋊C2, Dic7⋊C4, C4⋊Dic7, C23.D7, C2×C56, C2×Dic14, C2×D28, C2×D28, C22×C28, C23×D7, C2.D56, C7×C22⋊C8, C2×C56⋊C2, C28.48D4, C22×D28, D28.31D4
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C22≀C2, C2×SD16, C8⋊C22, D28, C22×D7, C22⋊SD16, C56⋊C2, C2×D28, D4×D7, C22⋊D28, C2×C56⋊C2, C8⋊D14, D28.31D4

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

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

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

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

79 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 ··· 28L 28M ··· 28R 56A ··· 56X 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 2 2 28 28 28 28 2 2 4 56 56 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

79 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D7 SD16 D14 D14 D28 D28 C56⋊C2 C8⋊C22 D4×D7 C8⋊D14 kernel D28.31D4 C2.D56 C7×C22⋊C8 C2×C56⋊C2 C28.48D4 C22×D28 D28 C2×C28 C22×C14 C22⋊C8 C2×C14 C2×C8 C22×C4 C2×C4 C23 C22 C14 C4 C2 # reps 1 2 1 2 1 1 4 1 1 3 4 6 3 6 6 24 1 6 6

Matrix representation of D28.31D4 in GL4(𝔽113) generated by

 109 32 0 0 81 58 0 0 0 0 112 0 0 0 0 112
,
 109 32 0 0 109 4 0 0 0 0 1 0 0 0 1 112
,
 89 71 0 0 42 50 0 0 0 0 112 2 0 0 0 1
,
 112 0 0 0 0 112 0 0 0 0 1 0 0 0 1 112
`G:=sub<GL(4,GF(113))| [109,81,0,0,32,58,0,0,0,0,112,0,0,0,0,112],[109,109,0,0,32,4,0,0,0,0,1,1,0,0,0,112],[89,42,0,0,71,50,0,0,0,0,112,0,0,0,2,1],[112,0,0,0,0,112,0,0,0,0,1,1,0,0,0,112] >;`

D28.31D4 in GAP, Magma, Sage, TeX

`D_{28}._{31}D_4`
`% in TeX`

`G:=Group("D28.31D4");`
`// GroupNames label`

`G:=SmallGroup(448,265);`
`// by ID`

`G=gap.SmallGroup(448,265);`
`# by ID`

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

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

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