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## G = C42.664C23order 128 = 27

### 79th non-split extension by C42 of C23 acting via C23/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — C42.664C23
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8⋊C8 — C42.664C23
 Lower central C1 — C22 — C42 — C42.664C23
 Upper central C1 — C22 — C42 — C42.664C23
 Jennings C1 — C22 — C22 — C42 — C42.664C23

Generators and relations for C42.664C23
G = < a,b,c,d,e | a4=b4=c2=1, d2=ab2, e2=b, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=a-1c, ece-1=b-1c, ede-1=a2d >

Subgroups: 304 in 104 conjugacy classes, 36 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C8 [×6], C2×C4 [×3], C2×C4 [×2], D4 [×12], Q8 [×4], C23 [×2], C42, C4⋊C4 [×4], C2×C8 [×6], D8 [×4], SD16 [×4], C2×D4 [×6], C2×Q8 [×2], C4×C8 [×3], D4⋊C4 [×6], Q8⋊C4 [×2], C41D4 [×2], C4⋊Q8 [×2], C2×D8 [×2], C2×SD16 [×2], C8⋊C8, C4.4D8 [×3], C4.SD16, C85D4, C84D4, C42.664C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C41D4, C4○D8 [×2], C8⋊C22 [×4], C8.12D4, C83D4 [×2], C42.664C23

Character table of C42.664C23

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L size 1 1 1 1 16 16 2 2 2 2 2 2 16 16 4 4 4 4 4 4 4 4 4 4 4 4 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 0 0 -2 2 -2 -2 2 -2 0 0 0 0 0 0 2 0 0 -2 0 0 -2 2 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -2 -2 -2 2 -2 2 0 0 0 0 -2 -2 0 0 2 0 0 2 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 2 -2 2 -2 -2 -2 0 0 -2 2 0 0 0 2 0 0 -2 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 0 -2 -2 -2 2 -2 2 0 0 0 0 2 2 0 0 -2 0 0 -2 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 -2 2 -2 -2 2 -2 0 0 0 0 0 0 -2 0 0 2 0 0 2 -2 orthogonal lifted from D4 ρ14 2 2 2 2 0 0 2 -2 2 -2 -2 -2 0 0 2 -2 0 0 0 -2 0 0 2 0 0 0 orthogonal lifted from D4 ρ15 2 -2 2 -2 0 0 0 -2 0 0 2 0 0 0 -√2 -√2 √-2 -√-2 0 √2 -√-2 -2i √2 √-2 2i 0 complex lifted from C4○D8 ρ16 2 -2 2 -2 0 0 0 -2 0 0 2 0 0 0 √2 √2 -√-2 √-2 0 -√2 √-2 -2i -√2 -√-2 2i 0 complex lifted from C4○D8 ρ17 2 -2 2 -2 0 0 0 -2 0 0 2 0 0 0 √2 √2 √-2 -√-2 0 -√2 -√-2 2i -√2 √-2 -2i 0 complex lifted from C4○D8 ρ18 2 -2 2 -2 0 0 0 2 0 0 -2 0 0 0 √2 -√2 √-2 -√-2 -2i √2 √-2 0 -√2 -√-2 0 2i complex lifted from C4○D8 ρ19 2 -2 2 -2 0 0 0 2 0 0 -2 0 0 0 -√2 √2 √-2 -√-2 2i -√2 √-2 0 √2 -√-2 0 -2i complex lifted from C4○D8 ρ20 2 -2 2 -2 0 0 0 2 0 0 -2 0 0 0 -√2 √2 -√-2 √-2 -2i -√2 -√-2 0 √2 √-2 0 2i complex lifted from C4○D8 ρ21 2 -2 2 -2 0 0 0 -2 0 0 2 0 0 0 -√2 -√2 -√-2 √-2 0 √2 √-2 2i √2 -√-2 -2i 0 complex lifted from C4○D8 ρ22 2 -2 2 -2 0 0 0 2 0 0 -2 0 0 0 √2 -√2 -√-2 √-2 2i √2 -√-2 0 -√2 √-2 0 -2i complex lifted from C4○D8 ρ23 4 -4 -4 4 0 0 0 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ24 4 4 -4 -4 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ25 4 -4 -4 4 0 0 0 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ26 4 4 -4 -4 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22

Smallest permutation representation of C42.664C23
On 64 points
Generators in S64
```(1 18 5 22)(2 19 6 23)(3 20 7 24)(4 21 8 17)(9 45 13 41)(10 46 14 42)(11 47 15 43)(12 48 16 44)(25 51 29 55)(26 52 30 56)(27 53 31 49)(28 54 32 50)(33 62 37 58)(34 63 38 59)(35 64 39 60)(36 57 40 61)
(1 61 24 34)(2 62 17 35)(3 63 18 36)(4 64 19 37)(5 57 20 38)(6 58 21 39)(7 59 22 40)(8 60 23 33)(9 25 43 49)(10 26 44 50)(11 27 45 51)(12 28 46 52)(13 29 47 53)(14 30 48 54)(15 31 41 55)(16 32 42 56)
(2 19)(3 7)(4 17)(6 23)(8 21)(9 25)(10 52)(11 31)(12 50)(13 29)(14 56)(15 27)(16 54)(18 22)(26 46)(28 44)(30 42)(32 48)(33 39)(34 61)(35 37)(36 59)(38 57)(40 63)(41 51)(43 49)(45 55)(47 53)(58 60)(62 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 9 61 25 24 43 34 49)(2 14 62 30 17 48 35 54)(3 11 63 27 18 45 36 51)(4 16 64 32 19 42 37 56)(5 13 57 29 20 47 38 53)(6 10 58 26 21 44 39 50)(7 15 59 31 22 41 40 55)(8 12 60 28 23 46 33 52)```

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

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

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

Matrix representation of C42.664C23 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0
,
 16 15 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 1 0 0 0 0 0 16 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 5 5 0 6 0 0 12 5 11 0 0 0 6 0 12 12 0 0 0 6 5 12
,
 10 10 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 16 0 0 0

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,5,12,6,0,0,0,5,5,0,6,0,0,0,11,12,5,0,0,6,0,12,12],[10,12,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C42.664C23 in GAP, Magma, Sage, TeX

`C_4^2._{664}C_2^3`
`% in TeX`

`G:=Group("C4^2.664C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,449);`
`// by ID`

`G=gap.SmallGroup(128,449);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,512,422,387,100,1123,136,2804,172]);`
`// Polycyclic`

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

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