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## G = C4⋊C4.84D4order 128 = 27

### 39th non-split extension by C4⋊C4 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 — C22×C4 — C4⋊C4.84D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×D4 — C24.3C22 — C4⋊C4.84D4
 Lower central C1 — C2 — C22×C4 — C4⋊C4.84D4
 Upper central C1 — C23 — C2×C42 — C4⋊C4.84D4
 Jennings C1 — C2 — C2 — C22×C4 — C4⋊C4.84D4

Generators and relations for C4⋊C4.84D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b2, bab-1=cac-1=a-1, ad=da, cbc-1=ab, dbd-1=b-1, dcd-1=a2b2c-1 >

Subgroups: 352 in 147 conjugacy classes, 50 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×7], C22 [×3], C22 [×4], C22 [×10], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×17], D4 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×12], C2×C8 [×6], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C24, D4⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42.C2 [×4], C22×C8 [×2], C22×D4, C22.7C42, C22.4Q16 [×2], C24.3C22, C2×D4⋊C4 [×2], C2×C42.C2, C4⋊C4.84D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], C4○D8 [×2], C8⋊C22 [×2], C23⋊Q8, D4⋊D4 [×2], D4.Q8 [×2], C42.78C22, C42.29C22, C4⋊C4.84D4

Smallest permutation representation of C4⋊C4.84D4
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 34 8 44)(2 33 5 43)(3 36 6 42)(4 35 7 41)(9 54 21 47)(10 53 22 46)(11 56 23 45)(12 55 24 48)(13 37 64 32)(14 40 61 31)(15 39 62 30)(16 38 63 29)(17 60 26 50)(18 59 27 49)(19 58 28 52)(20 57 25 51)
(1 32 12 26)(2 31 9 25)(3 30 10 28)(4 29 11 27)(5 40 21 20)(6 39 22 19)(7 38 23 18)(8 37 24 17)(13 54 50 33)(14 53 51 36)(15 56 52 35)(16 55 49 34)(41 62 45 58)(42 61 46 57)(43 64 47 60)(44 63 48 59)
(1 19 6 26)(2 20 7 27)(3 17 8 28)(4 18 5 25)(9 40 23 29)(10 37 24 30)(11 38 21 31)(12 39 22 32)(13 48 62 53)(14 45 63 54)(15 46 64 55)(16 47 61 56)(33 51 41 59)(34 52 42 60)(35 49 43 57)(36 50 44 58)```

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + - + image C1 C2 C2 C2 C2 C2 D4 D4 Q8 C4○D4 C4○D8 C8⋊C22 kernel C4⋊C4.84D4 C22.7C42 C22.4Q16 C24.3C22 C2×D4⋊C4 C2×C42.C2 C4⋊C4 C22×C4 C2×D4 C2×C4 C22 C22 # reps 1 1 2 1 2 1 4 2 2 6 8 2

Matrix representation of C4⋊C4.84D4 in GL6(𝔽17)

 16 2 0 0 0 0 16 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 1
,
 0 10 0 0 0 0 5 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 4 9 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 16 15 0 0 0 0 1 1
,
 4 9 0 0 0 0 4 13 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 16 15 0 0 0 0 0 1

`G:=sub<GL(6,GF(17))| [16,16,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,5,0,0,0,0,10,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,9,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,1,0,0,0,0,15,1],[4,4,0,0,0,0,9,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,15,1] >;`

C4⋊C4.84D4 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{84}D_4`
`% in TeX`

`G:=Group("C4:C4.84D4");`
`// GroupNames label`

`G:=SmallGroup(128,757);`
`// by ID`

`G=gap.SmallGroup(128,757);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,512,422,387,2804,718,172]);`
`// Polycyclic`

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

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