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## G = C42.244D4order 128 = 27

### 226th 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 — C42.244D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×M4(2) — C4×M4(2) — C42.244D4
 Lower central C1 — C2 — C2×C4 — C42.244D4
 Upper central C1 — C22 — C2×C42 — C42.244D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.244D4

Generators and relations for C42.244D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2b2c3 >

Subgroups: 372 in 196 conjugacy classes, 92 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], D4 [×12], Q8 [×4], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×12], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C42.C2 [×4], C42.C2 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×C4○D4 [×2], C4×M4(2), C23.36D4 [×4], C42.78C22 [×4], C42.29C22 [×2], C42.30C22 [×2], C2×C42.C2, C22.26C24, C42.244D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, D8⋊C22 [×2], C42.244D4

Smallest permutation representation of C42.244D4
On 64 points
Generators in S64
```(1 64 51 16)(2 57 52 9)(3 58 53 10)(4 59 54 11)(5 60 55 12)(6 61 56 13)(7 62 49 14)(8 63 50 15)(17 33 31 46)(18 34 32 47)(19 35 25 48)(20 36 26 41)(21 37 27 42)(22 38 28 43)(23 39 29 44)(24 40 30 45)
(1 22 5 18)(2 19 6 23)(3 24 7 20)(4 21 8 17)(9 48 13 44)(10 45 14 41)(11 42 15 46)(12 47 16 43)(25 56 29 52)(26 53 30 49)(27 50 31 54)(28 55 32 51)(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 54 51 4)(2 3 52 53)(5 50 55 8)(6 7 56 49)(9 62 57 14)(10 13 58 61)(11 60 59 12)(15 64 63 16)(17 22 31 28)(18 27 32 21)(19 20 25 26)(23 24 29 30)(33 47 46 34)(35 45 48 40)(36 39 41 44)(37 43 42 38)```

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

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

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

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 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 D8⋊C22 kernel C42.244D4 C4×M4(2) C23.36D4 C42.78C22 C42.29C22 C42.30C22 C2×C42.C2 C22.26C24 C42 C22×C4 C2×C4 C2 # reps 1 1 4 4 2 2 1 1 2 2 8 4

Matrix representation of C42.244D4 in GL6(𝔽17)

 16 8 0 0 0 0 4 1 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 16 0 0 0 0 0 0 16 0 0
,
 4 2 0 0 0 0 1 13 0 0 0 0 0 0 15 2 2 15 0 0 15 15 2 2 0 0 2 15 2 15 0 0 2 2 2 2
,
 4 2 0 0 0 0 0 13 0 0 0 0 0 0 2 15 15 2 0 0 15 15 2 2 0 0 15 2 15 2 0 0 2 2 2 2

`G:=sub<GL(6,GF(17))| [16,4,0,0,0,0,8,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[4,1,0,0,0,0,2,13,0,0,0,0,0,0,15,15,2,2,0,0,2,15,15,2,0,0,2,2,2,2,0,0,15,2,15,2],[4,0,0,0,0,0,2,13,0,0,0,0,0,0,2,15,15,2,0,0,15,15,2,2,0,0,15,2,15,2,0,0,2,2,2,2] >;`

C42.244D4 in GAP, Magma, Sage, TeX

`C_4^2._{244}D_4`
`% in TeX`

`G:=Group("C4^2.244D4");`
`// GroupNames label`

`G:=SmallGroup(128,1874);`
`// by ID`

`G=gap.SmallGroup(128,1874);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,680,758,100,1018,521,248,2804,172,4037,124]);`
`// 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*b^2,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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