Copied to
clipboard

## G = C42.261C23order 128 = 27

### 122nd 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 — C42.261C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C4×C4○D4 — C42.261C23
 Lower central C1 — C22 — C42.261C23
 Upper central C1 — C2×C4 — C42.261C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.261C23

Generators and relations for C42.261C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1b2, ad=da, eae=ab2, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, ede=b2d >

Subgroups: 268 in 192 conjugacy classes, 132 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×8], C2×C4 [×2], C2×C4 [×14], C2×C4 [×10], 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 [×2], C8⋊C4 [×6], 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×C8⋊C4, C4×M4(2), (C22×C8)⋊C2 [×2], C42.6C22 [×2], C42.6C4 [×4], C42.7C22 [×4], C4×C4○D4, C42.261C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, Q8○M4(2) [×2], C42.261C23

Smallest permutation representation of C42.261C23
On 64 points
Generators in S64
```(1 33 55 18)(2 23 56 38)(3 35 49 20)(4 17 50 40)(5 37 51 22)(6 19 52 34)(7 39 53 24)(8 21 54 36)(9 44 28 58)(10 63 29 41)(11 46 30 60)(12 57 31 43)(13 48 32 62)(14 59 25 45)(15 42 26 64)(16 61 27 47)
(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 10)(2 30)(3 12)(4 32)(5 14)(6 26)(7 16)(8 28)(9 54)(11 56)(13 50)(15 52)(17 62)(18 41)(19 64)(20 43)(21 58)(22 45)(23 60)(24 47)(25 51)(27 53)(29 55)(31 49)(33 63)(34 42)(35 57)(36 44)(37 59)(38 46)(39 61)(40 48)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(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)| (1,33,55,18)(2,23,56,38)(3,35,49,20)(4,17,50,40)(5,37,51,22)(6,19,52,34)(7,39,53,24)(8,21,54,36)(9,44,28,58)(10,63,29,41)(11,46,30,60)(12,57,31,43)(13,48,32,62)(14,59,25,45)(15,42,26,64)(16,61,27,47), (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), (1,10)(2,30)(3,12)(4,32)(5,14)(6,26)(7,16)(8,28)(9,54)(11,56)(13,50)(15,52)(17,62)(18,41)(19,64)(20,43)(21,58)(22,45)(23,60)(24,47)(25,51)(27,53)(29,55)(31,49)(33,63)(34,42)(35,57)(36,44)(37,59)(38,46)(39,61)(40,48), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(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( (1,33,55,18)(2,23,56,38)(3,35,49,20)(4,17,50,40)(5,37,51,22)(6,19,52,34)(7,39,53,24)(8,21,54,36)(9,44,28,58)(10,63,29,41)(11,46,30,60)(12,57,31,43)(13,48,32,62)(14,59,25,45)(15,42,26,64)(16,61,27,47), (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), (1,10)(2,30)(3,12)(4,32)(5,14)(6,26)(7,16)(8,28)(9,54)(11,56)(13,50)(15,52)(17,62)(18,41)(19,64)(20,43)(21,58)(22,45)(23,60)(24,47)(25,51)(27,53)(29,55)(31,49)(33,63)(34,42)(35,57)(36,44)(37,59)(38,46)(39,61)(40,48), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(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([(1,33,55,18),(2,23,56,38),(3,35,49,20),(4,17,50,40),(5,37,51,22),(6,19,52,34),(7,39,53,24),(8,21,54,36),(9,44,28,58),(10,63,29,41),(11,46,30,60),(12,57,31,43),(13,48,32,62),(14,59,25,45),(15,42,26,64),(16,61,27,47)], [(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)], [(1,10),(2,30),(3,12),(4,32),(5,14),(6,26),(7,16),(8,28),(9,54),(11,56),(13,50),(15,52),(17,62),(18,41),(19,64),(20,43),(21,58),(22,45),(23,60),(24,47),(25,51),(27,53),(29,55),(31,49),(33,63),(34,42),(35,57),(36,44),(37,59),(38,46),(39,61),(40,48)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,46),(18,47),(19,48),(20,41),(21,42),(22,43),(23,44),(24,45),(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)])`

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4○D4 Q8○M4(2) kernel C42.261C23 C2×C8⋊C4 C4×M4(2) (C22×C8)⋊C2 C42.6C22 C42.6C4 C42.7C22 C4×C4○D4 C4×D4 C4×Q8 C2×C4 C2 # reps 1 1 1 2 2 4 4 1 12 4 8 4

Matrix representation of C42.261C23 in GL6(𝔽17)

 13 0 0 0 0 0 6 4 0 0 0 0 0 0 0 0 0 13 0 0 0 0 4 0 0 0 0 4 0 0 0 0 13 0 0 0
,
 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
,
 14 13 0 0 0 0 11 3 0 0 0 0 0 0 0 10 0 7 0 0 10 0 7 0 0 0 0 7 0 7 0 0 7 0 7 0
,
 16 0 0 0 0 0 10 1 0 0 0 0 0 0 0 13 0 0 0 0 4 0 0 0 0 0 0 0 0 13 0 0 0 0 4 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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

C42.261C23 in GAP, Magma, Sage, TeX

`C_4^2._{261}C_2^3`
`% in TeX`

`G:=Group("C4^2.261C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1655);`
`// by ID`

`G=gap.SmallGroup(128,1655);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,100,1018,124]);`
`// 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*b^2,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`

׿
×
𝔽