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## G = C42.165D4order 128 = 27

### 147th non-split extension by C42 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C42.165D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4×D4 — C42.165D4
 Lower central C1 — C23 — C42.165D4
 Upper central C1 — C23 — C42.165D4
 Jennings C1 — C23 — C42.165D4

Generators and relations for C42.165D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 548 in 302 conjugacy classes, 112 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×16], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×14], C2×C4 [×40], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], C42 [×4], C42 [×4], C22⋊C4 [×20], C4⋊C4 [×20], C22×C4 [×3], C22×C4 [×10], C22×C4 [×12], C2×D4 [×6], C2×Q8 [×4], C2×Q8 [×4], C24 [×2], C2.C42 [×8], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×2], C2×C4⋊C4 [×8], C4×D4 [×4], C4×Q8 [×4], C22⋊Q8 [×8], C23×C4 [×2], C22×D4, C22×Q8, C23.7Q8 [×2], C428C4, C24.C22 [×4], C23.10D4 [×2], C23.81C23 [×2], C2×C4×D4, C2×C4×Q8, C2×C22⋊Q8 [×2], C42.165D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2- 1+4 [×2], C2×C4⋊D4, C22.19C24, C23.38C23, Q85D4 [×2], C22.46C24 [×2], C42.165D4

Smallest permutation representation of C42.165D4
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 11 14 25)(2 12 15 26)(3 9 16 27)(4 10 13 28)(5 39 22 30)(6 40 23 31)(7 37 24 32)(8 38 21 29)(17 36 62 51)(18 33 63 52)(19 34 64 49)(20 35 61 50)(41 53 60 46)(42 54 57 47)(43 55 58 48)(44 56 59 45)
(1 43 21 36)(2 57 22 50)(3 41 23 34)(4 59 24 52)(5 35 15 42)(6 49 16 60)(7 33 13 44)(8 51 14 58)(9 46 31 19)(10 56 32 63)(11 48 29 17)(12 54 30 61)(18 28 45 37)(20 26 47 39)(25 55 38 62)(27 53 40 64)
(1 57 14 42)(2 43 15 58)(3 59 16 44)(4 41 13 60)(5 51 22 36)(6 33 23 52)(7 49 24 34)(8 35 21 50)(9 56 27 45)(10 46 28 53)(11 54 25 47)(12 48 26 55)(17 39 62 30)(18 31 63 40)(19 37 64 32)(20 29 61 38)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4V 4W 4X 4Y 4Z order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 4 4 4 size 1 1 ··· 1 4 4 4 4 2 ··· 2 4 ··· 4 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 C4○D4 2- 1+4 kernel C42.165D4 C23.7Q8 C42⋊8C4 C24.C22 C23.10D4 C23.81C23 C2×C4×D4 C2×C4×Q8 C2×C22⋊Q8 C42 C2×Q8 C2×C4 C23 C22 # reps 1 2 1 4 2 2 1 1 2 4 4 4 8 2

Matrix representation of C42.165D4 in GL6(𝔽5)

 1 3 0 0 0 0 1 4 0 0 0 0 0 0 0 4 0 0 0 0 1 0 0 0 0 0 0 0 4 0 0 0 0 0 4 1
,
 1 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 2 0 0 0 0 0 2 3
,
 2 0 0 0 0 0 2 3 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 4 2 0 0 0 0 4 1
,
 3 4 0 0 0 0 3 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 3 0 0 0 0 1 4

`G:=sub<GL(6,GF(5))| [1,1,0,0,0,0,3,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,4,4,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,2,0,0,0,0,0,3],[2,2,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,4,0,0,0,0,2,1],[3,3,0,0,0,0,4,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,3,4] >;`

C42.165D4 in GAP, Magma, Sage, TeX

`C_4^2._{165}D_4`
`% in TeX`

`G:=Group("C4^2.165D4");`
`// GroupNames label`

`G:=SmallGroup(128,1268);`
`// by ID`

`G=gap.SmallGroup(128,1268);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,723,675,80]);`
`// Polycyclic`

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

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