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## G = C42.56C23order 128 = 27

### 56th non-split extension by C42 of C23 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.56C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — Q8⋊5D4 — C42.56C23
 Lower central C1 — C2 — C2×C4 — C42.56C23
 Upper central C1 — C22 — C4×D4 — C42.56C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.56C23

Generators and relations for C42.56C23
G = < a,b,c,d,e | a4=b4=1, c2=a2b2, d2=e2=b2, ab=ba, cac-1=eae-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2c, ede-1=b2d >

Subgroups: 400 in 196 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×12], C8 [×4], C2×C4 [×5], C2×C4 [×16], D4 [×11], Q8 [×2], Q8 [×6], C23 [×2], C23 [×2], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×5], C4⋊C4 [×3], C2×C8 [×4], C2×C8, M4(2), D8, SD16 [×3], C22×C4 [×2], C22×C4 [×3], C2×D4 [×3], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×3], C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×6], C4⋊C8, C4.Q8 [×2], C2.D8, C42⋊C2 [×2], C42⋊C2, C4×D4 [×2], C4×D4, C4×Q8, C4⋊D4 [×3], C4⋊D4 [×2], C22⋊Q8, C22⋊Q8, C4.4D4, C4.4D4 [×3], C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16 [×2], C22×Q8, C2×C4○D4, C23.24D4, C23.38D4, C89D4, SD16⋊C4, Q8⋊D4, D4⋊D4, D4.2D4, C88D4, C82D4, Q8⋊Q8, C23.19D4, C23.20D4, C42.28C22, Q85D4, C22.49C24, C42.56C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C22×D4, C2×C4○D4, 2+ 1+4, D45D4, D8⋊C22, D4○SD16, C42.56C23

Character table of C42.56C23

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 8A 8B 8C 8D 8E 8F size 1 1 1 1 4 4 8 8 2 2 2 2 4 4 4 4 4 4 4 8 8 8 8 4 4 4 4 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 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 -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 -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 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 -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 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 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 -1 1 1 linear of order 2 ρ9 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 1 -1 1 linear of order 2 ρ10 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 -1 -1 -1 linear of order 2 ρ11 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 -1 1 -1 linear of order 2 ρ12 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 1 1 1 linear of order 2 ρ13 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 -1 1 1 linear of order 2 ρ14 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 1 1 -1 linear of order 2 ρ15 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 1 -1 -1 linear of order 2 ρ16 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 -1 -1 1 linear of order 2 ρ17 2 2 2 2 2 -2 0 0 -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 ρ18 2 2 2 2 -2 -2 0 0 -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 ρ19 2 2 2 2 2 2 0 0 -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 ρ20 2 2 2 2 -2 2 0 0 -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 ρ21 2 -2 2 -2 0 0 0 0 2 -2 0 0 0 0 -2 2i 0 -2i 2 0 0 0 0 2i 0 0 -2i 0 0 complex lifted from C4○D4 ρ22 2 -2 2 -2 0 0 0 0 2 -2 0 0 0 0 2 2i 0 -2i -2 0 0 0 0 -2i 0 0 2i 0 0 complex lifted from C4○D4 ρ23 2 -2 2 -2 0 0 0 0 2 -2 0 0 0 0 2 -2i 0 2i -2 0 0 0 0 2i 0 0 -2i 0 0 complex lifted from C4○D4 ρ24 2 -2 2 -2 0 0 0 0 2 -2 0 0 0 0 -2 -2i 0 2i 2 0 0 0 0 -2i 0 0 2i 0 0 complex lifted from C4○D4 ρ25 4 -4 4 -4 0 0 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ26 4 -4 -4 4 0 0 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D8⋊C22 ρ27 4 -4 -4 4 0 0 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D8⋊C22 ρ28 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√-2 2√-2 0 0 0 complex lifted from D4○SD16 ρ29 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√-2 -2√-2 0 0 0 complex lifted from D4○SD16

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

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

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

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

Matrix representation of C42.56C23 in GL6(𝔽17)

 1 1 0 0 0 0 15 16 0 0 0 0 0 0 10 0 1 0 0 0 0 10 0 1 0 0 1 0 7 0 0 0 0 1 0 7
,
 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
,
 4 0 0 0 0 0 9 13 0 0 0 0 0 0 0 0 12 5 0 0 0 0 5 5 0 0 5 12 0 0 0 0 12 12 0 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 16 0 0
,
 16 16 0 0 0 0 0 1 0 0 0 0 0 0 16 0 10 0 0 0 0 16 0 10 0 0 10 0 1 0 0 0 0 10 0 1

`G:=sub<GL(6,GF(17))| [1,15,0,0,0,0,1,16,0,0,0,0,0,0,10,0,1,0,0,0,0,10,0,1,0,0,1,0,7,0,0,0,0,1,0,7],[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],[4,9,0,0,0,0,0,13,0,0,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,12,5,0,0,0,0,5,5,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,16,1,0,0,0,0,0,0,16,0,10,0,0,0,0,16,0,10,0,0,10,0,1,0,0,0,0,10,0,1] >;`

C42.56C23 in GAP, Magma, Sage, TeX

`C_4^2._{56}C_2^3`
`% in TeX`

`G:=Group("C4^2.56C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,2053);`
`// by ID`

`G=gap.SmallGroup(128,2053);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,723,352,346,248,4037,1027,124]);`
`// Polycyclic`

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

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