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## G = C42.290C23order 128 = 27

### 151st non-split extension by C42 of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42.290C23
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×C8 — C2×C4×C8 — C42.290C23
 Lower central C1 — C22 — C42.290C23
 Upper central C1 — C42 — C42.290C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.290C23

Generators and relations for C42.290C23
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=b-1, ab=ba, cac=a-1, dad-1=ab2, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=a2c, ede=b2d >

Subgroups: 276 in 206 conjugacy classes, 140 normal (36 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×10], C2×C4 [×16], D4 [×6], Q8 [×2], C23, C23 [×2], C42 [×4], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×8], C22×C4 [×3], C22×C4 [×6], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C4×C8 [×2], C4×C8 [×4], C8⋊C4 [×4], C22⋊C8 [×6], C4⋊C8 [×6], C2×C42, C2×C42 [×2], C42⋊C2, C42⋊C2 [×2], C4×D4 [×2], C4×D4 [×4], C4×Q8 [×2], C22×C8 [×2], C2×M4(2) [×4], C2×C4○D4, C2×C4×C8, C4×M4(2) [×2], C42.12C4, C42.12C4 [×2], C89D4 [×4], C86D4 [×2], C84Q8 [×2], C4×C4○D4, C42.290C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×4], C22×C4 [×14], C4○D4 [×4], C24, C2×M4(2) [×6], C8○D4 [×2], C23×C4, C2×C4○D4 [×2], C4×C4○D4, C22×M4(2), C2×C8○D4, C42.290C23

Smallest permutation representation of C42.290C23
On 64 points
Generators in S64
```(1 38 25 9)(2 35 26 14)(3 40 27 11)(4 37 28 16)(5 34 29 13)(6 39 30 10)(7 36 31 15)(8 33 32 12)(17 47 55 60)(18 44 56 57)(19 41 49 62)(20 46 50 59)(21 43 51 64)(22 48 52 61)(23 45 53 58)(24 42 54 63)
(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 44)(2 45)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 56)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 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 5)(3 7)(9 13)(11 15)(17 55)(18 52)(19 49)(20 54)(21 51)(22 56)(23 53)(24 50)(25 29)(27 31)(34 38)(36 40)(41 62)(42 59)(43 64)(44 61)(45 58)(46 63)(47 60)(48 57)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4L 4M ··· 4R 4S ··· 4X 8A ··· 8P 8Q ··· 8X order 1 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 C4○D4 M4(2) C8○D4 kernel C42.290C23 C2×C4×C8 C4×M4(2) C42.12C4 C8⋊9D4 C8⋊6D4 C8⋊4Q8 C4×C4○D4 C42⋊C2 C4×D4 C4×Q8 C2×C4○D4 C8 C2×C4 C4 # reps 1 1 2 3 4 2 2 1 6 6 2 2 8 8 8

Matrix representation of C42.290C23 in GL4(𝔽17) generated by

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

C42.290C23 in GAP, Magma, Sage, TeX

`C_4^2._{290}C_2^3`
`% in TeX`

`G:=Group("C4^2.290C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1697);`
`// by ID`

`G=gap.SmallGroup(128,1697);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,184,80,124]);`
`// Polycyclic`

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

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