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## G = C56.C23order 448 = 26·7

### 6th non-split extension by C56 of C23 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C56.C23
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C4○D28 — Q8.10D14 — C56.C23
 Lower central C7 — C14 — C28 — C56.C23
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for C56.C23
G = < a,b,c,d | a56=b2=1, c2=d2=a28, bab=a13, cac-1=a15, dad-1=a43, bc=cb, dbd-1=a28b, cd=dc >

Subgroups: 1388 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C23, D7, C14, C14, C2×C8, M4(2), M4(2), D8, SD16, SD16, Q16, Q16, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, C8.C22, 2+ 1+4, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, D28, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C7×Q8, C22×D7, D4○SD16, C8×D7, C8⋊D7, C56⋊C2, D56, C2×C7⋊C8, D4⋊D7, D4.D7, Q8⋊D7, Q8⋊D7, C7⋊Q16, C7×M4(2), C7×SD16, C7×Q16, C2×D28, C2×D28, C4○D28, C4○D28, D4×D7, D4×D7, Q8×D7, Q8×D7, Q82D7, Q82D7, Q8×C14, C7×C4○D4, D28.C4, C8⋊D14, D7×SD16, D56⋊C2, Q16⋊D7, Q8.D14, C2×Q8⋊D7, D4.8D14, C7×C8.C22, Q8.10D14, D48D14, C56.C23
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D4○SD16, D4×D7, C23×D7, C2×D4×D7, C56.C23

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

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

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

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

55 conjugacy classes

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

55 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 D14 D14 D14 D4○SD16 D4×D7 D4×D7 C56.C23 kernel C56.C23 D28.C4 C8⋊D14 D7×SD16 D56⋊C2 Q16⋊D7 Q8.D14 C2×Q8⋊D7 D4.8D14 C7×C8.C22 Q8.10D14 D4⋊8D14 Dic14 D28 C7⋊D4 C8.C22 M4(2) SD16 Q16 C2×Q8 C4○D4 C7 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 1 1 2 3 3 6 6 3 3 2 3 3 3

Matrix representation of C56.C23 in GL6(𝔽113)

 80 112 0 0 0 0 2 24 0 0 0 0 0 0 100 100 0 0 0 0 13 100 0 0 0 0 0 0 13 100 0 0 0 0 13 13
,
 8 80 0 0 0 0 43 105 0 0 0 0 0 0 0 0 13 100 0 0 0 0 100 100 0 0 100 13 0 0 0 0 13 13 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 112 0 0 0 0 0 0 112 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 100 100 0 0 0 0 100 13 0 0 0 0 0 0 100 100 0 0 0 0 100 13

`G:=sub<GL(6,GF(113))| [80,2,0,0,0,0,112,24,0,0,0,0,0,0,100,13,0,0,0,0,100,100,0,0,0,0,0,0,13,13,0,0,0,0,100,13],[8,43,0,0,0,0,80,105,0,0,0,0,0,0,0,0,100,13,0,0,0,0,13,13,0,0,13,100,0,0,0,0,100,100,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,100,100,0,0,0,0,100,13,0,0,0,0,0,0,100,100,0,0,0,0,100,13] >;`

C56.C23 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(448,1231);`
`// by ID`

`G=gap.SmallGroup(448,1231);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,184,570,185,438,235,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^56=b^2=1,c^2=d^2=a^28,b*a*b=a^13,c*a*c^-1=a^15,d*a*d^-1=a^43,b*c=c*b,d*b*d^-1=a^28*b,c*d=d*c>;`
`// generators/relations`

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