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## G = C42.243D4order 128 = 27

### 225th non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.243D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×M4(2) — C4×M4(2) — C42.243D4
 Lower central C1 — C2 — C2×C4 — C42.243D4
 Upper central C1 — C22 — C2×C42 — C42.243D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.243D4

Generators and relations for C42.243D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2b2c3 >

Subgroups: 420 in 212 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×18], D4 [×12], Q8 [×12], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×8], C4○D4 [×8], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C4⋊Q8 [×4], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8, C2×C4○D4 [×2], C4×M4(2), C23.36D4 [×4], C4.4D8 [×2], C4.SD16 [×2], C42.28C22 [×4], C2×C4⋊Q8, C22.26C24, C42.243D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C8⋊C22 [×2], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, C2×C8⋊C22, C2×C8.C22, C42.243D4

Smallest permutation representation of C42.243D4
On 64 points
Generators in S64
```(1 60 55 20)(2 61 56 21)(3 62 49 22)(4 63 50 23)(5 64 51 24)(6 57 52 17)(7 58 53 18)(8 59 54 19)(9 46 35 26)(10 47 36 27)(11 48 37 28)(12 41 38 29)(13 42 39 30)(14 43 40 31)(15 44 33 32)(16 45 34 25)
(1 45 5 41)(2 42 6 46)(3 47 7 43)(4 44 8 48)(9 21 13 17)(10 18 14 22)(11 23 15 19)(12 20 16 24)(25 51 29 55)(26 56 30 52)(27 53 31 49)(28 50 32 54)(33 59 37 63)(34 64 38 60)(35 61 39 57)(36 58 40 62)
(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 50 55 4)(2 3 56 49)(5 54 51 8)(6 7 52 53)(9 36 35 10)(11 34 37 16)(12 15 38 33)(13 40 39 14)(17 58 57 18)(19 64 59 24)(20 23 60 63)(21 62 61 22)(25 48 45 28)(26 27 46 47)(29 44 41 32)(30 31 42 43)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4H 4I 4J 4K ··· 4P 8A ··· 8H order 1 2 2 2 2 2 2 2 4 ··· 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 8 8 2 ··· 2 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 C8⋊C22 C8.C22 kernel C42.243D4 C4×M4(2) C23.36D4 C4.4D8 C4.SD16 C42.28C22 C2×C4⋊Q8 C22.26C24 C42 C22×C4 C2×C4 C4 C4 # reps 1 1 4 2 2 4 1 1 2 2 8 2 2

Matrix representation of C42.243D4 in GL6(𝔽17)

 1 8 0 0 0 0 4 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 1 16 1 15 0 0 16 0 0 0 0 0 16 1 0 1
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 16 0 0 1 16 0 16 0 0 0 16 0 0
,
 4 0 0 0 0 0 16 13 0 0 0 0 0 0 0 0 0 16 0 0 1 0 1 16 0 0 16 1 0 1 0 0 16 0 0 0

`G:=sub<GL(6,GF(17))| [1,4,0,0,0,0,8,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,16,16,0,0,0,16,0,1,0,0,1,1,0,0,0,0,0,15,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,16,0,0,0,1,0,0,0,0,1,16,16,0],[4,16,0,0,0,0,0,13,0,0,0,0,0,0,0,1,16,16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16,16,1,0] >;`

C42.243D4 in GAP, Magma, Sage, TeX

`C_4^2._{243}D_4`
`% in TeX`

`G:=Group("C4^2.243D4");`
`// GroupNames label`

`G:=SmallGroup(128,1873);`
`// by ID`

`G=gap.SmallGroup(128,1873);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,100,1018,521,248,2804,172,4037,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^3>;`
`// generators/relations`

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