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## G = C112⋊C4order 448 = 26·7

### 2nd semidirect product of C112 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C112⋊C4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C56 — C56⋊C4 — C112⋊C4
 Lower central C7 — C28 — C112⋊C4
 Upper central C1 — C4 — M5(2)

Generators and relations for C112⋊C4
G = < a,b | a112=b4=1, bab-1=a13 >

Smallest permutation representation of C112⋊C4
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)
(2 70 58 14)(3 27)(4 96 60 40)(5 53)(6 10 62 66)(7 79)(8 36 64 92)(9 105)(11 19)(12 88 68 32)(13 45)(15 71)(16 28 72 84)(17 97)(18 54 74 110)(20 80 76 24)(21 37)(22 106 78 50)(23 63)(25 89)(26 46 82 102)(30 98 86 42)(31 55)(33 81)(34 38 90 94)(35 107)(39 47)(41 73)(43 99)(44 56 100 112)(48 108 104 52)(49 65)(51 91)(59 83)(61 109)(67 75)(69 101)(77 93)(87 111)(95 103)```

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

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

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

82 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 8C 8D 8E 8F 14A 14B 14C 14D 14E 14F 16A 16B 16C 16D 16E 16F 16G 16H 28A ··· 28F 28G 28H 28I 56A ··· 56L 56M ··· 56R 112A ··· 112X order 1 2 2 4 4 4 4 4 7 7 7 8 8 8 8 8 8 14 14 14 14 14 14 16 16 16 16 16 16 16 16 28 ··· 28 28 28 28 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 2 1 1 2 28 28 2 2 2 2 2 2 2 28 28 2 2 2 4 4 4 4 4 4 4 28 28 28 28 2 ··· 2 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - + image C1 C2 C2 C2 C4 C4 C4 C4 D7 M4(2) M4(2) Dic7 D14 C4×D7 C4×D7 C8⋊D7 C8⋊D7 C16⋊C4 C112⋊C4 kernel C112⋊C4 C28.C8 C56⋊C4 C7×M5(2) C7⋊C16 C112 C2×C7⋊C8 C4×Dic7 M5(2) C28 C2×C14 C16 C2×C8 C8 C2×C4 C4 C22 C7 C1 # reps 1 1 1 1 4 4 2 2 3 2 2 6 3 6 6 12 12 2 12

Matrix representation of C112⋊C4 in GL4(𝔽113) generated by

 0 0 88 88 0 0 25 34 110 3 0 0 110 8 0 0
,
 1 0 0 0 79 112 0 0 0 0 98 0 0 0 58 15
`G:=sub<GL(4,GF(113))| [0,0,110,110,0,0,3,8,88,25,0,0,88,34,0,0],[1,79,0,0,0,112,0,0,0,0,98,58,0,0,0,15] >;`

C112⋊C4 in GAP, Magma, Sage, TeX

`C_{112}\rtimes C_4`
`% in TeX`

`G:=Group("C112:C4");`
`// GroupNames label`

`G:=SmallGroup(448,69);`
`// by ID`

`G=gap.SmallGroup(448,69);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,387,1123,80,102,18822]);`
`// Polycyclic`

`G:=Group<a,b|a^112=b^4=1,b*a*b^-1=a^13>;`
`// generators/relations`

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