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## G = C42.73D4order 128 = 27

### 55th 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 — C42.73D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C22.26C24 — C42.73D4
 Lower central C1 — C22 — C2×C4 — C42.73D4
 Upper central C1 — C22 — C2×C42 — C42.73D4
 Jennings C1 — C22 — C22 — C42 — C42.73D4

Generators and relations for C42.73D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=a2b-1c3 >

Subgroups: 276 in 119 conjugacy classes, 44 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×7], C22, C22 [×9], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×10], Q8 [×2], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×4], C8⋊C4 [×4], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C2×C4○D4 [×2], C42.C22 [×4], C42.6C4 [×2], C22.26C24, C42.73D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.D4 [×2], C2×C22⋊C4, C2×C4.D4, C42⋊C22 [×2], C42.73D4

Character table of C42.73D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 4 8 8 2 2 2 2 2 2 4 4 4 8 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 linear of order 2 ρ5 1 1 1 1 -1 -1 1 1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 linear of order 2 ρ6 1 1 1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 1 1 -1 -1 -1 1 -1 linear of order 2 ρ7 1 1 1 1 -1 -1 1 1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 linear of order 2 ρ8 1 1 1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 1 -1 1 linear of order 2 ρ9 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 -i i -i i -i i i -i linear of order 4 ρ10 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 i i -i i i -i -i -i linear of order 4 ρ11 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 i -i i -i i -i -i i linear of order 4 ρ12 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 -i -i i -i -i i i i linear of order 4 ρ13 1 1 1 1 1 -1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 -1 i i -i -i -i i -i i linear of order 4 ρ14 1 1 1 1 1 1 -1 -1 1 -1 1 -1 -1 -1 1 -1 -1 1 -i i -i -i i -i i i linear of order 4 ρ15 1 1 1 1 1 -1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 -1 -i -i i i i -i i -i linear of order 4 ρ16 1 1 1 1 1 1 -1 -1 1 -1 1 -1 -1 -1 1 -1 -1 1 i -i i i -i i -i -i linear of order 4 ρ17 2 2 2 2 2 0 0 2 -2 -2 -2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 0 0 2 2 -2 2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 -2 0 0 -2 2 2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 0 0 -2 -2 2 -2 2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 -4 4 0 0 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ22 4 -4 -4 4 0 0 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ23 4 -4 4 -4 0 0 0 -4i 0 0 0 0 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊C22 ρ24 4 -4 4 -4 0 0 0 4i 0 0 0 0 -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊C22 ρ25 4 4 -4 -4 0 0 0 0 0 4i 0 -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊C22 ρ26 4 4 -4 -4 0 0 0 0 0 -4i 0 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42⋊C22

Smallest permutation representation of C42.73D4
On 64 points
Generators in S64
```(1 35 56 48)(2 41 49 36)(3 37 50 42)(4 43 51 38)(5 39 52 44)(6 45 53 40)(7 33 54 46)(8 47 55 34)(9 32 20 58)(10 59 21 25)(11 26 22 60)(12 61 23 27)(13 28 24 62)(14 63 17 29)(15 30 18 64)(16 57 19 31)
(1 27 52 57)(2 32 53 62)(3 29 54 59)(4 26 55 64)(5 31 56 61)(6 28 49 58)(7 25 50 63)(8 30 51 60)(9 45 24 36)(10 42 17 33)(11 47 18 38)(12 44 19 35)(13 41 20 40)(14 46 21 37)(15 43 22 34)(16 48 23 39)
(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 30 27 51 52 60 57 8)(2 54 32 59 53 3 62 29)(4 5 26 31 55 56 64 61)(6 50 28 63 49 7 58 25)(9 21 45 37 24 14 36 46)(10 40 42 13 17 41 33 20)(11 16 47 48 18 23 38 39)(12 43 44 22 19 34 35 15)```

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

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

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

Matrix representation of C42.73D4 in GL8(𝔽17)

 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0
,
 16 15 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 15 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13
,
 13 12 0 0 0 0 0 0 4 4 0 0 0 0 0 0 0 0 4 5 0 0 0 0 0 0 13 13 0 0 0 0 0 0 0 0 11 6 8 9 0 0 0 0 11 11 8 8 0 0 0 0 8 9 6 11 0 0 0 0 8 8 6 6
,
 13 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 5 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 11 6 8 9 0 0 0 0 6 6 9 9 0 0 0 0 8 9 6 11 0 0 0 0 9 9 11 11

`G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13],[13,4,0,0,0,0,0,0,12,4,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,5,13,0,0,0,0,0,0,0,0,11,11,8,8,0,0,0,0,6,11,9,8,0,0,0,0,8,8,6,6,0,0,0,0,9,8,11,6],[13,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,5,16,0,0,0,0,0,0,0,0,11,6,8,9,0,0,0,0,6,6,9,9,0,0,0,0,8,9,6,11,0,0,0,0,9,9,11,11] >;`

C42.73D4 in GAP, Magma, Sage, TeX

`C_4^2._{73}D_4`
`% in TeX`

`G:=Group("C4^2.73D4");`
`// GroupNames label`

`G:=SmallGroup(128,268);`
`// by ID`

`G=gap.SmallGroup(128,268);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,352,1123,1018,248,1971,102]);`
`// Polycyclic`

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

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