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## G = C28.8C42order 448 = 26·7

### 1st non-split extension by C28 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C28.8C42
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C22×C28 — C23.21D14 — C28.8C42
 Lower central C7 — C14 — C28 — C28.8C42
 Upper central C1 — C2×C4 — C22×C4 — C2×C42

Generators and relations for C28.8C42
G = < a,b,c | a28=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=a7b >

Subgroups: 356 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2×C42, C42⋊C2, C2×M4(2), C7⋊C8, C2×Dic7, C2×C28, C2×C28, C22×C14, C426C4, C2×C7⋊C8, C4.Dic7, C4.Dic7, C4×Dic7, C4⋊Dic7, C23.D7, C4×C28, C4×C28, C22×C28, C22×C28, C2×C4.Dic7, C23.21D14, C2×C4×C28, C28.8C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, C4≀C2, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C426C4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, Dic14⋊C4, C14.C42, C28.8C42

Smallest permutation representation of C28.8C42
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)
(1 103 34 68)(2 102 35 67)(3 101 36 66)(4 100 37 65)(5 99 38 64)(6 98 39 63)(7 97 40 62)(8 96 41 61)(9 95 42 60)(10 94 43 59)(11 93 44 58)(12 92 45 57)(13 91 46 84)(14 90 47 83)(15 89 48 82)(16 88 49 81)(17 87 50 80)(18 86 51 79)(19 85 52 78)(20 112 53 77)(21 111 54 76)(22 110 55 75)(23 109 56 74)(24 108 29 73)(25 107 30 72)(26 106 31 71)(27 105 32 70)(28 104 33 69)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(57 85 71 99)(58 86 72 100)(59 87 73 101)(60 88 74 102)(61 89 75 103)(62 90 76 104)(63 91 77 105)(64 92 78 106)(65 93 79 107)(66 94 80 108)(67 95 81 109)(68 96 82 110)(69 97 83 111)(70 98 84 112)```

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

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

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

124 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 7A 7B 7C 8A 8B 8C 8D 14A ··· 14U 28A ··· 28BT order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 28 28 28 28 2 2 2 28 28 28 28 2 ··· 2 2 ··· 2

124 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + - + - + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 D4 D7 Dic7 D14 C4≀C2 Dic14 C4×D7 D28 C7⋊D4 C7⋊D4 Dic14⋊C4 kernel C28.8C42 C2×C4.Dic7 C23.21D14 C2×C4×C28 C4.Dic7 C4⋊Dic7 C4×C28 C2×C28 C2×C28 C22×C14 C2×C42 C42 C22×C4 C14 C2×C4 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 1 1 1 4 4 4 2 1 1 3 6 3 8 6 12 6 6 6 48

Matrix representation of C28.8C42 in GL4(𝔽113) generated by

 98 0 0 0 66 15 0 0 0 0 64 0 0 0 0 83
,
 99 45 0 0 81 14 0 0 0 0 0 1 0 0 112 0
,
 1 0 0 0 10 98 0 0 0 0 112 0 0 0 0 1
`G:=sub<GL(4,GF(113))| [98,66,0,0,0,15,0,0,0,0,64,0,0,0,0,83],[99,81,0,0,45,14,0,0,0,0,0,112,0,0,1,0],[1,10,0,0,0,98,0,0,0,0,112,0,0,0,0,1] >;`

C28.8C42 in GAP, Magma, Sage, TeX

`C_{28}._8C_4^2`
`% in TeX`

`G:=Group("C28.8C4^2");`
`// GroupNames label`

`G:=SmallGroup(448,80);`
`// by ID`

`G=gap.SmallGroup(448,80);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,1123,1684,102,18822]);`
`// Polycyclic`

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

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