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G = C42.266C23order 128 = 27

127th 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 — C42.266C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C8○D4 — C42.266C23
 Lower central C1 — C22 — C42.266C23
 Upper central C1 — C2×C4 — C42.266C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.266C23

Generators and relations for C42.266C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2b, ab=ba, cac-1=a-1, dad-1=ab2, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a2c, ece=b2c, de=ed >

Subgroups: 364 in 242 conjugacy classes, 140 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×14], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×12], C2×C4 [×10], D4 [×16], Q8 [×4], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×12], C2×C8 [×10], M4(2) [×12], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], C8⋊C4 [×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×C8⋊C4, (C22×C8)⋊C2 [×2], C4⋊M4(2), C89D4 [×8], C22.26C24, C2×C8○D4 [×2], C42.266C23
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], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, Q8○M4(2) [×2], C42.266C23

Smallest permutation representation of C42.266C23
On 64 points
Generators in S64
```(1 45 32 24)(2 42 25 21)(3 47 26 18)(4 44 27 23)(5 41 28 20)(6 46 29 17)(7 43 30 22)(8 48 31 19)(9 37 51 58)(10 34 52 63)(11 39 53 60)(12 36 54 57)(13 33 55 62)(14 38 56 59)(15 35 49 64)(16 40 50 61)
(1 26 5 30)(2 27 6 31)(3 28 7 32)(4 29 8 25)(9 53 13 49)(10 54 14 50)(11 55 15 51)(12 56 16 52)(17 48 21 44)(18 41 22 45)(19 42 23 46)(20 43 24 47)(33 64 37 60)(34 57 38 61)(35 58 39 62)(36 59 40 63)
(1 15 5 11)(2 50 6 54)(3 9 7 13)(4 52 8 56)(10 31 14 27)(12 25 16 29)(17 57 21 61)(18 37 22 33)(19 59 23 63)(20 39 24 35)(26 51 30 55)(28 53 32 49)(34 48 38 44)(36 42 40 46)(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 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)(41 60)(42 61)(43 62)(44 63)(45 64)(46 57)(47 58)(48 59)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G ··· 4N 8A ··· 8H 8I ··· 8T order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 4 4 4 4 1 1 1 1 2 2 4 ··· 4 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 Q8○M4(2) kernel C42.266C23 C2×C8⋊C4 (C22×C8)⋊C2 C4⋊M4(2) C8⋊9D4 C22.26C24 C2×C8○D4 C4⋊D4 C4.4D4 C4⋊1D4 C4⋊Q8 C2×C8 C2×C4 C2 # reps 1 1 2 1 8 1 2 8 4 2 2 4 4 4

Matrix representation of C42.266C23 in GL6(𝔽17)

 0 4 0 0 0 0 4 0 0 0 0 0 0 0 8 6 6 14 0 0 0 8 3 0 0 0 0 1 9 0 0 0 16 0 6 9
,
 16 0 0 0 0 0 0 16 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 13
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 1 11 0 0 16 0 6 1 0 0 0 0 0 16 0 0 0 0 1 0
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 16 12 12 7 0 0 0 16 10 0 0 0 0 2 1 0 0 0 15 0 12 1
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 11 0 0 1 0 11 0 0 0 0 0 0 16 0 0 0 0 16 0

`G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,4,0,0,0,0,0,0,0,8,0,0,16,0,0,6,8,1,0,0,0,6,3,9,6,0,0,14,0,0,9],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,1,6,0,1,0,0,11,1,16,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,15,0,0,12,16,2,0,0,0,12,10,1,12,0,0,7,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,11,0,16,0,0,11,0,16,0] >;`

C42.266C23 in GAP, Magma, Sage, TeX

`C_4^2._{266}C_2^3`
`% in TeX`

`G:=Group("C4^2.266C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1664);`
`// by ID`

`G=gap.SmallGroup(128,1664);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,184,2019,1018,124]);`
`// Polycyclic`

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

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