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## G = C42.681C23order 128 = 27

### 96th non-split extension by C42 of C23 acting via C23/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42.681C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C8○D4 — C42.681C23
 Lower central C1 — C22 — C42.681C23
 Upper central C1 — C2×C8 — C42.681C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.681C23

Generators and relations for C42.681C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2b-1, ab=ba, cac-1=a-1b2, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a2b2c, ece=b2c, de=ed >

Subgroups: 364 in 250 conjugacy classes, 144 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C4 [×6], C22, C22 [×2], C22 [×14], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×12], C2×C4 [×12], D4 [×20], Q8 [×4], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×10], C2×C8 [×10], M4(2) [×12], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], C4×C8 [×4], C22⋊C8 [×8], C4⋊C8 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C22×C8 [×2], C22×C8 [×4], C2×M4(2) [×6], C8○D4 [×8], C2×C4○D4 [×2], C2×C4×C8, (C22×C8)⋊C2 [×2], C4⋊M4(2), C8×D4 [×4], C86D4 [×4], C22.26C24, C2×C8○D4 [×2], C42.681C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C8○D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C8○D4 [×2], C42.681C23

Smallest permutation representation of C42.681C23
On 64 points
Generators in S64
```(1 41 32 20)(2 42 25 21)(3 43 26 22)(4 44 27 23)(5 45 28 24)(6 46 29 17)(7 47 30 18)(8 48 31 19)(9 54 57 40)(10 55 58 33)(11 56 59 34)(12 49 60 35)(13 50 61 36)(14 51 62 37)(15 52 63 38)(16 53 64 39)
(1 30 5 26)(2 31 6 27)(3 32 7 28)(4 25 8 29)(9 63 13 59)(10 64 14 60)(11 57 15 61)(12 58 16 62)(17 44 21 48)(18 45 22 41)(19 46 23 42)(20 47 24 43)(33 53 37 49)(34 54 38 50)(35 55 39 51)(36 56 40 52)
(1 39 5 35)(2 50 6 54)(3 33 7 37)(4 52 8 56)(9 46 13 42)(10 22 14 18)(11 48 15 44)(12 24 16 20)(17 61 21 57)(19 63 23 59)(25 36 29 40)(26 55 30 51)(27 38 31 34)(28 49 32 53)(41 60 45 64)(43 62 47 58)
(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 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(41 60)(42 61)(43 62)(44 63)(45 64)(46 57)(47 58)(48 59)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 8A ··· 8H 8I ··· 8T 8U ··· 8AB order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 8 ··· 8 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 4 4 4 4 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 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 D4 C4○D4 C8○D4 kernel C42.681C23 C2×C4×C8 (C22×C8)⋊C2 C4⋊M4(2) C8×D4 C8⋊6D4 C22.26C24 C2×C8○D4 C4⋊D4 C4.4D4 C4⋊1D4 C4⋊Q8 C2×C8 C2×C4 C4 # reps 1 1 2 1 4 4 1 2 8 4 2 2 4 4 16

Matrix representation of C42.681C23 in GL4(𝔽17) generated by

 4 0 0 0 0 4 0 0 0 0 0 1 0 0 1 0
,
 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 1 2 0 0 16 16 0 0 0 0 0 1 0 0 16 0
,
 2 0 0 0 0 2 0 0 0 0 0 8 0 0 8 0
,
 16 15 0 0 0 1 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,16,0,0,2,16,0,0,0,0,0,16,0,0,1,0],[2,0,0,0,0,2,0,0,0,0,0,8,0,0,8,0],[16,0,0,0,15,1,0,0,0,0,0,1,0,0,1,0] >;`

C42.681C23 in GAP, Magma, Sage, TeX

`C_4^2._{681}C_2^3`
`% in TeX`

`G:=Group("C4^2.681C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1663);`
`// by ID`

`G=gap.SmallGroup(128,1663);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2019,1018,124]);`
`// Polycyclic`

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

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