Copied to
clipboard

## G = C23.19C42order 128 = 27

### 1st non-split extension by C23 of C42 acting via C42/C2×C4=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C23.19C42
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C2.C42 — C23.19C42
 Lower central C1 — C2 — C22 — C23.19C42
 Upper central C1 — C23 — C23×C4 — C23.19C42
 Jennings C1 — C2 — C22 — C23×C4 — C23.19C42

Generators and relations for C23.19C42
G = < a,b,c,d,e | a2=b2=c2=e4=1, d4=c, dad-1=ab=ba, ac=ca, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=abd >

Subgroups: 280 in 132 conjugacy classes, 52 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×8], C22 [×5], C22 [×6], C22 [×12], C8 [×4], C2×C4 [×4], C2×C4 [×24], C23 [×3], C23 [×4], C23 [×4], C2×C8 [×8], C22×C4 [×2], C22×C4 [×8], C22×C4 [×10], C24, C2.C42 [×2], C22⋊C8 [×4], C22⋊C8 [×2], C22×C8 [×2], C23×C4, C23×C4 [×2], C2×C2.C42, C2×C22⋊C8 [×2], C23.19C42
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C23⋊C4 [×2], C4.D4, C4.10D4, C4⋊C8 [×2], C23⋊C8 [×2], C22.M4(2) [×2], C22.7C42, C23.9D4, C22.C42, C23.19C42

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

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4P 8A ··· 8P order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 4 4 type + + + + - + + - image C1 C2 C2 C4 C4 C8 D4 Q8 M4(2) C23⋊C4 C4.D4 C4.10D4 kernel C23.19C42 C2×C2.C42 C2×C22⋊C8 C22⋊C8 C23×C4 C22×C4 C22×C4 C22×C4 C23 C22 C22 C22 # reps 1 1 2 8 4 16 3 1 4 2 1 1

Matrix representation of C23.19C42 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 5 8 0 16
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 3 15 0 0 0 0 0 0 5 14 0 0 0 0 0 0 0 0 2 3 0 0 0 0 0 0 3 15 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 6 13 10 1 0 0 0 0 16 15 0 0 0 0 0 0 1 5 0 4
,
 13 0 0 0 0 0 0 0 5 4 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 6 6 0 0 0 0 0 0 14 11 0 0 0 0 0 0 2 10 15 6 0 0 0 0 15 0 8 2

`G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,1,0,8,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[3,5,0,0,0,0,0,0,15,14,0,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,3,15,0,0,0,0,0,0,0,0,0,6,16,1,0,0,0,0,0,13,15,5,0,0,0,0,1,10,0,0,0,0,0,0,0,1,0,4],[13,5,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,6,14,2,15,0,0,0,0,6,11,10,0,0,0,0,0,0,0,15,8,0,0,0,0,0,0,6,2] >;`

C23.19C42 in GAP, Magma, Sage, TeX

`C_2^3._{19}C_4^2`
`% in TeX`

`G:=Group("C2^3.19C4^2");`
`// GroupNames label`

`G:=SmallGroup(128,12);`
`// by ID`

`G=gap.SmallGroup(128,12);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,570,248]);`
`// Polycyclic`

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

׿
×
𝔽