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## G = C43⋊15C2order 128 = 27

### 15th semidirect product of C43 and C2 acting faithfully

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C4315C2, C4241D4, C23.767C24, C24.128C23, C41(C41D4), C22.473(C22×D4), (C2×C42).1098C22, (C22×C4).1487C23, (C22×D4).317C22, (C2×C41D4)⋊13C2, (C2×C4).837(C2×D4), C2.18(C2×C41D4), SmallGroup(128,1599)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C43⋊15C2
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C43⋊15C2
 Lower central C1 — C23 — C43⋊15C2
 Upper central C1 — C23 — C43⋊15C2
 Jennings C1 — C23 — C43⋊15C2

Generators and relations for C4315C2
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 1476 in 636 conjugacy classes, 180 normal (4 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, D4, C23, C23, C42, C22×C4, C2×D4, C24, C2×C42, C41D4, C22×D4, C43, C2×C41D4, C4315C2
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C41D4, C22×D4, C2×C41D4, C4315C2

Smallest permutation representation of C4315C2
On 64 points
Generators in S64
```(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)
(1 17 9 45)(2 18 10 46)(3 19 11 47)(4 20 12 48)(5 60 39 30)(6 57 40 31)(7 58 37 32)(8 59 38 29)(13 49 41 21)(14 50 42 22)(15 51 43 23)(16 52 44 24)(25 63 55 34)(26 64 56 35)(27 61 53 36)(28 62 54 33)
(1 57 53 41)(2 58 54 42)(3 59 55 43)(4 60 56 44)(5 64 52 48)(6 61 49 45)(7 62 50 46)(8 63 51 47)(9 31 27 13)(10 32 28 14)(11 29 25 15)(12 30 26 16)(17 40 36 21)(18 37 33 22)(19 38 34 23)(20 39 35 24)
(1 49)(2 52)(3 51)(4 50)(5 54)(6 53)(7 56)(8 55)(9 21)(10 24)(11 23)(12 22)(13 17)(14 20)(15 19)(16 18)(25 38)(26 37)(27 40)(28 39)(29 34)(30 33)(31 36)(32 35)(41 45)(42 48)(43 47)(44 46)(57 61)(58 64)(59 63)(60 62)```

`G:=sub<Sym(64)| (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), (1,17,9,45)(2,18,10,46)(3,19,11,47)(4,20,12,48)(5,60,39,30)(6,57,40,31)(7,58,37,32)(8,59,38,29)(13,49,41,21)(14,50,42,22)(15,51,43,23)(16,52,44,24)(25,63,55,34)(26,64,56,35)(27,61,53,36)(28,62,54,33), (1,57,53,41)(2,58,54,42)(3,59,55,43)(4,60,56,44)(5,64,52,48)(6,61,49,45)(7,62,50,46)(8,63,51,47)(9,31,27,13)(10,32,28,14)(11,29,25,15)(12,30,26,16)(17,40,36,21)(18,37,33,22)(19,38,34,23)(20,39,35,24), (1,49)(2,52)(3,51)(4,50)(5,54)(6,53)(7,56)(8,55)(9,21)(10,24)(11,23)(12,22)(13,17)(14,20)(15,19)(16,18)(25,38)(26,37)(27,40)(28,39)(29,34)(30,33)(31,36)(32,35)(41,45)(42,48)(43,47)(44,46)(57,61)(58,64)(59,63)(60,62)>;`

`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), (1,17,9,45)(2,18,10,46)(3,19,11,47)(4,20,12,48)(5,60,39,30)(6,57,40,31)(7,58,37,32)(8,59,38,29)(13,49,41,21)(14,50,42,22)(15,51,43,23)(16,52,44,24)(25,63,55,34)(26,64,56,35)(27,61,53,36)(28,62,54,33), (1,57,53,41)(2,58,54,42)(3,59,55,43)(4,60,56,44)(5,64,52,48)(6,61,49,45)(7,62,50,46)(8,63,51,47)(9,31,27,13)(10,32,28,14)(11,29,25,15)(12,30,26,16)(17,40,36,21)(18,37,33,22)(19,38,34,23)(20,39,35,24), (1,49)(2,52)(3,51)(4,50)(5,54)(6,53)(7,56)(8,55)(9,21)(10,24)(11,23)(12,22)(13,17)(14,20)(15,19)(16,18)(25,38)(26,37)(27,40)(28,39)(29,34)(30,33)(31,36)(32,35)(41,45)(42,48)(43,47)(44,46)(57,61)(58,64)(59,63)(60,62) );`

`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)], [(1,17,9,45),(2,18,10,46),(3,19,11,47),(4,20,12,48),(5,60,39,30),(6,57,40,31),(7,58,37,32),(8,59,38,29),(13,49,41,21),(14,50,42,22),(15,51,43,23),(16,52,44,24),(25,63,55,34),(26,64,56,35),(27,61,53,36),(28,62,54,33)], [(1,57,53,41),(2,58,54,42),(3,59,55,43),(4,60,56,44),(5,64,52,48),(6,61,49,45),(7,62,50,46),(8,63,51,47),(9,31,27,13),(10,32,28,14),(11,29,25,15),(12,30,26,16),(17,40,36,21),(18,37,33,22),(19,38,34,23),(20,39,35,24)], [(1,49),(2,52),(3,51),(4,50),(5,54),(6,53),(7,56),(8,55),(9,21),(10,24),(11,23),(12,22),(13,17),(14,20),(15,19),(16,18),(25,38),(26,37),(27,40),(28,39),(29,34),(30,33),(31,36),(32,35),(41,45),(42,48),(43,47),(44,46),(57,61),(58,64),(59,63),(60,62)]])`

44 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4AB order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 2 type + + + + image C1 C2 C2 D4 kernel C43⋊15C2 C43 C2×C4⋊1D4 C42 # reps 1 1 14 28

Matrix representation of C4315C2 in GL6(ℤ)

 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 2 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -2 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 0 0 0 0 1 0
,
 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 -2 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 0 0 0 0 -1 0
,
 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1

`G:=sub<GL(6,Integers())| [0,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,2,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,-2,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,0],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,-2,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1] >;`

C4315C2 in GAP, Magma, Sage, TeX

`C_4^3\rtimes_{15}C_2`
`% in TeX`

`G:=Group("C4^3:15C2");`
`// GroupNames label`

`G:=SmallGroup(128,1599);`
`// by ID`

`G=gap.SmallGroup(128,1599);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,184,2019,248]);`
`// Polycyclic`

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

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