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## G = C4⋊C4.98D4order 128 = 27

### 53rd 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⋊C4.98D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×M4(2) — C2×C4.10D4 — C4⋊C4.98D4
 Lower central C1 — C2 — C22×C4 — C4⋊C4.98D4
 Upper central C1 — C22 — C22×C4 — C4⋊C4.98D4
 Jennings C1 — C2 — C2 — C22×C4 — C4⋊C4.98D4

Generators and relations for C4⋊C4.98D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b-1, dbd-1=ab, dcd-1=a2c3 >

Subgroups: 248 in 128 conjugacy classes, 42 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×2], C8 [×5], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×12], C23, C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×5], M4(2) [×6], Q16 [×8], C22×C4, C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×10], C4.10D4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C42⋊C2, C22×C8, C2×M4(2), C2×M4(2) [×2], C2×Q16 [×6], C22×Q8 [×2], C4.C42, C2×C4.10D4 [×2], C23.38D4 [×2], C42.6C22, C22×Q16, C4⋊C4.98D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C23.10D4, D4.5D4 [×2], C4⋊C4.98D4

Character table of C4⋊C4.98D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 1 1 2 2 2 2 2 2 8 8 8 8 8 8 4 4 4 4 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 2 -2 -2 2 -2 2 -2 -2 2 2 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 2 -2 2 -2 -2 2 0 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 -2 2 -2 -2 2 2 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 -2 2 -2 2 0 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -2 -2 2 -2 2 -2 0 0 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 -2 2 -2 2 -2 0 0 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 -2 2 2 -2 2 -2 -2 2 0 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ16 2 2 2 2 -2 -2 -2 2 -2 2 0 0 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 -2 -2 2 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 -2i 0 0 0 2i 0 complex lifted from C4○D4 ρ18 2 -2 -2 2 -2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 -2i 0 0 0 2i complex lifted from C4○D4 ρ19 2 -2 -2 2 -2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 2i 0 0 0 -2i complex lifted from C4○D4 ρ20 2 2 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 complex lifted from C4○D4 ρ21 2 2 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 2i -2i 0 0 complex lifted from C4○D4 ρ22 2 -2 -2 2 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 2i 0 0 0 -2i 0 complex lifted from C4○D4 ρ23 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 2√2 0 0 -2√2 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2 ρ24 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2 ρ25 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2 ρ26 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 0 0 2√2 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2

Smallest permutation representation of C4⋊C4.98D4
On 64 points
Generators in S64
```(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 55 53 51)(50 56 54 52)(57 59 61 63)(58 60 62 64)
(1 19 63 55)(2 52 64 24)(3 21 57 49)(4 54 58 18)(5 23 59 51)(6 56 60 20)(7 17 61 53)(8 50 62 22)(9 36 27 46)(10 43 28 33)(11 38 29 48)(12 45 30 35)(13 40 31 42)(14 47 32 37)(15 34 25 44)(16 41 26 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 36 5 40)(2 35 6 39)(3 34 7 38)(4 33 8 37)(9 53 13 49)(10 52 14 56)(11 51 15 55)(12 50 16 54)(17 31 21 27)(18 30 22 26)(19 29 23 25)(20 28 24 32)(41 64 45 60)(42 63 46 59)(43 62 47 58)(44 61 48 57)```

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

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

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

Matrix representation of C4⋊C4.98D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 2 0 0 0 0 16 1 0 0 0 0 0 0 1 15 0 0 0 0 1 16
,
 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 14 6 0 0 11 6 0 0 0 0 14 0 0 0
,
 16 11 0 0 0 0 6 1 0 0 0 0 0 0 0 6 0 0 0 0 14 6 0 0 0 0 0 0 0 6 0 0 0 0 14 6
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 6 2 0 0 0 0 7 11 0 0 0 0 0 0 6 2 0 0 0 0 7 11

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,11,14,0,0,0,0,6,0,0,0,0,14,0,0,0,0,6,6,0,0],[16,6,0,0,0,0,11,1,0,0,0,0,0,0,0,14,0,0,0,0,6,6,0,0,0,0,0,0,0,14,0,0,0,0,6,6],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,6,7,0,0,0,0,2,11,0,0,0,0,0,0,6,7,0,0,0,0,2,11] >;`

C4⋊C4.98D4 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{98}D_4`
`% in TeX`

`G:=Group("C4:C4.98D4");`
`// GroupNames label`

`G:=SmallGroup(128,779);`
`// by ID`

`G=gap.SmallGroup(128,779);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,456,422,387,58,2019,1018,248,718,1027]);`
`// Polycyclic`

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

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