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## G = C56.49C23order 448 = 26·7

### 42nd non-split extension by C56 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C56.49C23
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C2×C4×D7 — D7×C4○D4 — C56.49C23
 Lower central C7 — C14 — C56.49C23
 Upper central C1 — C4 — C8○D4

Generators and relations for C56.49C23
G = < a,b,c,d | a56=b2=c2=d2=1, bab=a13, cac=a29, ad=da, bc=cb, bd=db, dcd=a28c >

Subgroups: 956 in 258 conjugacy classes, 147 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, D14, C2×C14, C2×M4(2), C8○D4, C8○D4, C2×C4○D4, C7⋊C8, C7⋊C8, C56, C56, Dic14, C4×D7, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×Q8, C22×D7, Q8○M4(2), C8×D7, C8⋊D7, C8⋊D7, C2×C7⋊C8, C4.Dic7, C2×C56, C7×M4(2), C2×C4×D7, C4○D28, D4×D7, D42D7, Q8×D7, Q82D7, C7×C4○D4, C2×C8⋊D7, D28.2C4, D7×M4(2), D28.C4, Q8.Dic7, C7×C8○D4, D7×C4○D4, C56.49C23
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C24, D14, C23×C4, C4×D7, C22×D7, Q8○M4(2), C2×C4×D7, C23×D7, D7×C22×C4, C56.49C23

Smallest permutation representation of C56.49C23
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 14)(3 27)(4 40)(5 53)(6 10)(7 23)(8 36)(9 49)(11 19)(12 32)(13 45)(16 28)(17 41)(18 54)(20 24)(21 37)(22 50)(25 33)(26 46)(30 42)(31 55)(34 38)(35 51)(39 47)(44 56)(48 52)(57 61)(58 74)(59 87)(60 100)(62 70)(63 83)(64 96)(65 109)(67 79)(68 92)(69 105)(71 75)(72 88)(73 101)(76 84)(77 97)(78 110)(81 93)(82 106)(85 89)(86 102)(90 98)(91 111)(95 107)(99 103)(104 112)
(1 94)(2 67)(3 96)(4 69)(5 98)(6 71)(7 100)(8 73)(9 102)(10 75)(11 104)(12 77)(13 106)(14 79)(15 108)(16 81)(17 110)(18 83)(19 112)(20 85)(21 58)(22 87)(23 60)(24 89)(25 62)(26 91)(27 64)(28 93)(29 66)(30 95)(31 68)(32 97)(33 70)(34 99)(35 72)(36 101)(37 74)(38 103)(39 76)(40 105)(41 78)(42 107)(43 80)(44 109)(45 82)(46 111)(47 84)(48 57)(49 86)(50 59)(51 88)(52 61)(53 90)(54 63)(55 92)(56 65)
(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(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), (2,14)(3,27)(4,40)(5,53)(6,10)(7,23)(8,36)(9,49)(11,19)(12,32)(13,45)(16,28)(17,41)(18,54)(20,24)(21,37)(22,50)(25,33)(26,46)(30,42)(31,55)(34,38)(35,51)(39,47)(44,56)(48,52)(57,61)(58,74)(59,87)(60,100)(62,70)(63,83)(64,96)(65,109)(67,79)(68,92)(69,105)(71,75)(72,88)(73,101)(76,84)(77,97)(78,110)(81,93)(82,106)(85,89)(86,102)(90,98)(91,111)(95,107)(99,103)(104,112), (1,94)(2,67)(3,96)(4,69)(5,98)(6,71)(7,100)(8,73)(9,102)(10,75)(11,104)(12,77)(13,106)(14,79)(15,108)(16,81)(17,110)(18,83)(19,112)(20,85)(21,58)(22,87)(23,60)(24,89)(25,62)(26,91)(27,64)(28,93)(29,66)(30,95)(31,68)(32,97)(33,70)(34,99)(35,72)(36,101)(37,74)(38,103)(39,76)(40,105)(41,78)(42,107)(43,80)(44,109)(45,82)(46,111)(47,84)(48,57)(49,86)(50,59)(51,88)(52,61)(53,90)(54,63)(55,92)(56,65), (57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(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), (2,14)(3,27)(4,40)(5,53)(6,10)(7,23)(8,36)(9,49)(11,19)(12,32)(13,45)(16,28)(17,41)(18,54)(20,24)(21,37)(22,50)(25,33)(26,46)(30,42)(31,55)(34,38)(35,51)(39,47)(44,56)(48,52)(57,61)(58,74)(59,87)(60,100)(62,70)(63,83)(64,96)(65,109)(67,79)(68,92)(69,105)(71,75)(72,88)(73,101)(76,84)(77,97)(78,110)(81,93)(82,106)(85,89)(86,102)(90,98)(91,111)(95,107)(99,103)(104,112), (1,94)(2,67)(3,96)(4,69)(5,98)(6,71)(7,100)(8,73)(9,102)(10,75)(11,104)(12,77)(13,106)(14,79)(15,108)(16,81)(17,110)(18,83)(19,112)(20,85)(21,58)(22,87)(23,60)(24,89)(25,62)(26,91)(27,64)(28,93)(29,66)(30,95)(31,68)(32,97)(33,70)(34,99)(35,72)(36,101)(37,74)(38,103)(39,76)(40,105)(41,78)(42,107)(43,80)(44,109)(45,82)(46,111)(47,84)(48,57)(49,86)(50,59)(51,88)(52,61)(53,90)(54,63)(55,92)(56,65), (57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(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)], [(2,14),(3,27),(4,40),(5,53),(6,10),(7,23),(8,36),(9,49),(11,19),(12,32),(13,45),(16,28),(17,41),(18,54),(20,24),(21,37),(22,50),(25,33),(26,46),(30,42),(31,55),(34,38),(35,51),(39,47),(44,56),(48,52),(57,61),(58,74),(59,87),(60,100),(62,70),(63,83),(64,96),(65,109),(67,79),(68,92),(69,105),(71,75),(72,88),(73,101),(76,84),(77,97),(78,110),(81,93),(82,106),(85,89),(86,102),(90,98),(91,111),(95,107),(99,103),(104,112)], [(1,94),(2,67),(3,96),(4,69),(5,98),(6,71),(7,100),(8,73),(9,102),(10,75),(11,104),(12,77),(13,106),(14,79),(15,108),(16,81),(17,110),(18,83),(19,112),(20,85),(21,58),(22,87),(23,60),(24,89),(25,62),(26,91),(27,64),(28,93),(29,66),(30,95),(31,68),(32,97),(33,70),(34,99),(35,72),(36,101),(37,74),(38,103),(39,76),(40,105),(41,78),(42,107),(43,80),(44,109),(45,82),(46,111),(47,84),(48,57),(49,86),(50,59),(51,88),(52,61),(53,90),(54,63),(55,92),(56,65)], [(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)]])`

94 conjugacy classes

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

94 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D7 D14 D14 D14 C4×D7 C4×D7 Q8○M4(2) C56.49C23 kernel C56.49C23 C2×C8⋊D7 D28.2C4 D7×M4(2) D28.C4 Q8.Dic7 C7×C8○D4 D7×C4○D4 D4×D7 D4⋊2D7 Q8×D7 Q8⋊2D7 C8○D4 C2×C8 M4(2) C4○D4 D4 Q8 C7 C1 # reps 1 3 3 3 3 1 1 1 6 6 2 2 3 9 9 3 18 6 2 12

Matrix representation of C56.49C23 in GL4(𝔽113) generated by

 76 49 0 0 64 105 0 0 0 0 37 64 0 0 49 8
,
 1 0 0 0 79 112 0 0 0 0 1 0 0 0 79 112
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
,
 1 0 0 0 0 1 0 0 0 0 112 0 0 0 0 112
`G:=sub<GL(4,GF(113))| [76,64,0,0,49,105,0,0,0,0,37,49,0,0,64,8],[1,79,0,0,0,112,0,0,0,0,1,79,0,0,0,112],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112] >;`

C56.49C23 in GAP, Magma, Sage, TeX

`C_{56}._{49}C_2^3`
`% in TeX`

`G:=Group("C56.49C2^3");`
`// GroupNames label`

`G:=SmallGroup(448,1203);`
`// by ID`

`G=gap.SmallGroup(448,1203);`
`# by ID`

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

`G:=Group<a,b,c,d|a^56=b^2=c^2=d^2=1,b*a*b=a^13,c*a*c=a^29,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^28*c>;`
`// generators/relations`

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