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## G = C42.449D4order 128 = 27

### 82nd non-split extension by C42 of D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.449D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C4×C4○D4 — C42.449D4
 Lower central C1 — C2 — C2×C4 — C42.449D4
 Upper central C1 — C22 — C2×C42 — C42.449D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.449D4

Generators and relations for C42.449D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2c3 >

Subgroups: 324 in 192 conjugacy classes, 100 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×21], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×2], C42 [×2], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×6], C4⋊C4 [×12], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C42.C2 [×4], C42.C2 [×2], C2×M4(2) [×2], C2×C4○D4, C23.36D4 [×2], C4⋊M4(2), M4(2)⋊C4 [×2], D4.Q8 [×4], Q8.Q8 [×4], C4×C4○D4, C2×C42.C2, C42.449D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, D8⋊C22 [×2], C42.449D4

Smallest permutation representation of C42.449D4
On 64 points
Generators in S64
```(1 15 45 63)(2 60 46 12)(3 9 47 57)(4 62 48 14)(5 11 41 59)(6 64 42 16)(7 13 43 61)(8 58 44 10)(17 55 35 26)(18 31 36 52)(19 49 37 28)(20 25 38 54)(21 51 39 30)(22 27 40 56)(23 53 33 32)(24 29 34 50)
(1 26 5 30)(2 31 6 27)(3 28 7 32)(4 25 8 29)(9 19 13 23)(10 24 14 20)(11 21 15 17)(12 18 16 22)(33 57 37 61)(34 62 38 58)(35 59 39 63)(36 64 40 60)(41 51 45 55)(42 56 46 52)(43 53 47 49)(44 50 48 54)
(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 48 41 8)(2 7 42 47)(3 46 43 6)(4 5 44 45)(9 64 61 12)(10 11 62 63)(13 60 57 16)(14 15 58 59)(17 38 39 24)(18 23 40 37)(19 36 33 22)(20 21 34 35)(25 26 50 51)(27 32 52 49)(28 56 53 31)(29 30 54 55)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4H 4I ··· 4P 4Q 4R 4S 4T 8A 8B 8C 8D order 1 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 4 4 4 4 8 8 8 8 size 1 1 1 1 2 2 4 4 2 ··· 2 4 ··· 4 8 8 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 Q8 C4○D4 D8⋊C22 kernel C42.449D4 C23.36D4 C4⋊M4(2) M4(2)⋊C4 D4.Q8 Q8.Q8 C4×C4○D4 C2×C42.C2 C42 C22×C4 C4○D4 C2×C4 C2 # reps 1 2 1 2 4 4 1 1 2 2 4 4 4

Matrix representation of C42.449D4 in GL6(𝔽17)

 4 15 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 4 0 0 0 0 0 0 4 0 0
,
 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 0 0 0 0 0 16 13 0 0 0 0 0 0 7 7 0 0 0 0 5 0 0 0 0 0 0 0 10 10 0 0 0 0 12 0
,
 13 0 0 0 0 0 1 4 0 0 0 0 0 0 7 7 0 0 0 0 5 10 0 0 0 0 0 0 10 10 0 0 0 0 12 7

`G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,15,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,13,0,0,0,0,0,0,13,0,0],[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],[4,16,0,0,0,0,0,13,0,0,0,0,0,0,7,5,0,0,0,0,7,0,0,0,0,0,0,0,10,12,0,0,0,0,10,0],[13,1,0,0,0,0,0,4,0,0,0,0,0,0,7,5,0,0,0,0,7,10,0,0,0,0,0,0,10,12,0,0,0,0,10,7] >;`

C42.449D4 in GAP, Magma, Sage, TeX

`C_4^2._{449}D_4`
`% in TeX`

`G:=Group("C4^2.449D4");`
`// GroupNames label`

`G:=SmallGroup(128,1812);`
`// by ID`

`G=gap.SmallGroup(128,1812);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,568,758,2019,248,4037,1027,124]);`
`// Polycyclic`

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

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