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## G = C42.100D4order 128 = 27

### 82nd 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 — C42.100D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×D4 — C2×C4×D4 — C42.100D4
 Lower central C1 — C2 — C2×C4 — C42.100D4
 Upper central C1 — C23 — C2×C42 — C42.100D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.100D4

Generators and relations for C42.100D4
G = < a,b,c,d | a4=b4=c4=1, d2=cbc-1=b-1, ab=ba, cac-1=dad-1=a-1b2, bd=db, dcd-1=b-1c-1 >

Subgroups: 388 in 178 conjugacy classes, 68 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C22 [×16], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×22], D4 [×4], D4 [×6], C23, C23 [×10], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×6], C22×C4 [×3], C22×C4 [×11], C2×D4 [×6], C2×D4 [×3], C24, C2.C42 [×2], D4⋊C4 [×4], C4⋊C8 [×2], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4 [×2], C4×D4 [×4], C4×D4 [×2], C22×C8 [×2], C23×C4, C22×D4, C22.4Q16 [×2], C428C4, C2×D4⋊C4 [×2], C2×C4⋊C8, C2×C4×D4, C42.100D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C4○D8 [×2], C8⋊C22 [×2], C23.7Q8, C23.24D4, C23.37D4, D4.2D4 [×2], D4.Q8 [×2], C42.100D4

Smallest permutation representation of C42.100D4
On 64 points
Generators in S64
```(1 59 51 47)(2 44 52 64)(3 61 53 41)(4 46 54 58)(5 63 55 43)(6 48 56 60)(7 57 49 45)(8 42 50 62)(9 38 17 29)(10 26 18 35)(11 40 19 31)(12 28 20 37)(13 34 21 25)(14 30 22 39)(15 36 23 27)(16 32 24 33)
(1 7 5 3)(2 8 6 4)(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 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 36 32 54)(2 53 25 35)(3 34 26 52)(4 51 27 33)(5 40 28 50)(6 49 29 39)(7 38 30 56)(8 55 31 37)(9 18 48 61)(10 60 41 17)(11 24 42 59)(12 58 43 23)(13 22 44 57)(14 64 45 21)(15 20 46 63)(16 62 47 19)
(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)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4N 4O 4P 4Q 4R 8A ··· 8H order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 4 4 4 4 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4 D4 D4 D4 Q8 C4○D4 C4○D8 C8⋊C22 kernel C42.100D4 C22.4Q16 C42⋊8C4 C2×D4⋊C4 C2×C4⋊C8 C2×C4×D4 C4×D4 C42 C22×C4 C2×D4 C2×D4 C2×C4 C22 C22 # reps 1 2 1 2 1 1 8 2 2 2 2 4 8 2

Matrix representation of C42.100D4 in GL6(𝔽17)

 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 16 13 0 0 0 0 9 1
,
 0 1 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 16 0 0 0 0 0 0 16
,
 14 3 0 0 0 0 3 3 0 0 0 0 0 0 5 12 0 0 0 0 12 12 0 0 0 0 0 0 0 3 0 0 0 0 11 0
,
 14 3 0 0 0 0 14 14 0 0 0 0 0 0 5 12 0 0 0 0 5 5 0 0 0 0 0 0 0 14 0 0 0 0 6 0

`G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,9,0,0,0,0,13,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[14,3,0,0,0,0,3,3,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,0,0,0,0,0,11,0,0,0,0,3,0],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,6,0,0,0,0,14,0] >;`

C42.100D4 in GAP, Magma, Sage, TeX

`C_4^2._{100}D_4`
`% in TeX`

`G:=Group("C4^2.100D4");`
`// GroupNames label`

`G:=SmallGroup(128,536);`
`// by ID`

`G=gap.SmallGroup(128,536);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,512,422,2804,718,172]);`
`// Polycyclic`

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

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