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## G = C42.260C23order 128 = 27

### 121st 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 — C42.260C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C4×C4○D4 — C42.260C23
 Lower central C1 — C22 — C42.260C23
 Upper central C1 — C42 — C42.260C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.260C23

Generators and relations for C42.260C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1, ad=da, eae=ab2, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, ede=b2d >

Subgroups: 268 in 200 conjugacy classes, 136 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×8], C2×C4 [×2], C2×C4 [×14], C2×C4 [×16], D4 [×6], Q8 [×2], C23, C23 [×2], C42 [×2], C42 [×8], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×8], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C4×C8 [×6], C8⋊C4 [×2], C22⋊C8 [×8], C4⋊C8 [×8], C2×C42, C2×C42 [×2], C42⋊C2, C42⋊C2 [×2], C4×D4 [×6], C4×Q8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×C4○D4, C2×C4×C8, C4×M4(2), (C22×C8)⋊C2 [×2], C42.6C22 [×2], C42.12C4 [×4], C42.7C22 [×4], C4×C4○D4, C42.260C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C8○D4 [×4], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, C2×C8○D4 [×2], C42.260C23

Smallest permutation representation of C42.260C23
On 64 points
Generators in S64
```(1 33 55 44)(2 45 56 34)(3 35 49 46)(4 47 50 36)(5 37 51 48)(6 41 52 38)(7 39 53 42)(8 43 54 40)(9 28 62 22)(10 23 63 29)(11 30 64 24)(12 17 57 31)(13 32 58 18)(14 19 59 25)(15 26 60 20)(16 21 61 27)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(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 23)(2 30)(3 17)(4 32)(5 19)(6 26)(7 21)(8 28)(9 40)(10 44)(11 34)(12 46)(13 36)(14 48)(15 38)(16 42)(18 50)(20 52)(22 54)(24 56)(25 51)(27 53)(29 55)(31 49)(33 63)(35 57)(37 59)(39 61)(41 60)(43 62)(45 64)(47 58)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 45)(10 46)(11 47)(12 48)(13 41)(14 42)(15 43)(16 44)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)```

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

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

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

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 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4○D4 C8○D4 kernel C42.260C23 C2×C4×C8 C4×M4(2) (C22×C8)⋊C2 C42.6C22 C42.12C4 C42.7C22 C4×C4○D4 C4×D4 C4×Q8 C2×C4 C4 # reps 1 1 1 2 2 4 4 1 12 4 8 16

Matrix representation of C42.260C23 in GL4(𝔽17) generated by

 0 16 0 0 1 0 0 0 0 0 0 4 0 0 13 0
,
 4 0 0 0 0 4 0 0 0 0 13 0 0 0 0 13
,
 0 2 0 0 2 0 0 0 0 0 9 0 0 0 0 9
,
 0 4 0 0 13 0 0 0 0 0 0 13 0 0 4 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,0,13,0,0,4,0],[4,0,0,0,0,4,0,0,0,0,13,0,0,0,0,13],[0,2,0,0,2,0,0,0,0,0,9,0,0,0,0,9],[0,13,0,0,4,0,0,0,0,0,0,4,0,0,13,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;`

C42.260C23 in GAP, Magma, Sage, TeX

`C_4^2._{260}C_2^3`
`% in TeX`

`G:=Group("C4^2.260C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1654);`
`// by ID`

`G=gap.SmallGroup(128,1654);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,1018,124]);`
`// Polycyclic`

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

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