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## G = C42.674C23order 128 = 27

### 89th 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 — C42.674C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C4×C4○D4 — C42.674C23
 Lower central C1 — C22 — C42.674C23
 Upper central C1 — C2×C4 — C42.674C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.674C23

Generators and relations for C42.674C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 316 in 242 conjugacy classes, 174 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×8], C4 [×7], C22, C22 [×6], C22 [×6], C8 [×8], C2×C4 [×3], C2×C4 [×21], C2×C4 [×9], D4 [×12], Q8 [×4], C23 [×3], C42, C42 [×9], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×12], M4(2) [×12], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C4⋊C8, C4⋊C8 [×15], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C22×C8 [×6], C2×M4(2) [×6], C8○D4 [×8], C2×C4○D4, C2×C4⋊C8 [×3], C4⋊M4(2) [×3], C42.6C22 [×6], C4×C4○D4, C2×C8○D4 [×2], C42.674C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C8○D4 [×2], C23×C4, C22×D4, C22×Q8, C22×C4⋊C4, C2×C8○D4, Q8○M4(2), C42.674C23

Smallest permutation representation of C42.674C23
On 64 points
Generators in S64
```(1 63 55 16)(2 9 56 64)(3 57 49 10)(4 11 50 58)(5 59 51 12)(6 13 52 60)(7 61 53 14)(8 15 54 62)(17 33 29 47)(18 48 30 34)(19 35 31 41)(20 42 32 36)(21 37 25 43)(22 44 26 38)(23 39 27 45)(24 46 28 40)
(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 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 49)(32 50)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(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)| (1,63,55,16)(2,9,56,64)(3,57,49,10)(4,11,50,58)(5,59,51,12)(6,13,52,60)(7,61,53,14)(8,15,54,62)(17,33,29,47)(18,48,30,34)(19,35,31,41)(20,42,32,36)(21,37,25,43)(22,44,26,38)(23,39,27,45)(24,46,28,40), (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), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(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( (1,63,55,16)(2,9,56,64)(3,57,49,10)(4,11,50,58)(5,59,51,12)(6,13,52,60)(7,61,53,14)(8,15,54,62)(17,33,29,47)(18,48,30,34)(19,35,31,41)(20,42,32,36)(21,37,25,43)(22,44,26,38)(23,39,27,45)(24,46,28,40), (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), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(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([(1,63,55,16),(2,9,56,64),(3,57,49,10),(4,11,50,58),(5,59,51,12),(6,13,52,60),(7,61,53,14),(8,15,54,62),(17,33,29,47),(18,48,30,34),(19,35,31,41),(20,42,32,36),(21,37,25,43),(22,44,26,38),(23,39,27,45),(24,46,28,40)], [(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)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,48),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,49),(32,50),(33,63),(34,64),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(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)])`

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 D4 Q8 C8○D4 Q8○M4(2) kernel C42.674C23 C2×C4⋊C8 C4⋊M4(2) C42.6C22 C4×C4○D4 C2×C8○D4 C4×D4 C4×Q8 C4○D4 C4○D4 C4 C2 # reps 1 3 3 6 1 2 12 4 4 4 8 2

Matrix representation of C42.674C23 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 13 0 0 0 9 4
,
 4 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 2 0 0 0 0 2 0 0 0 0 1 16 0 0 0 16
,
 13 9 0 0 4 4 0 0 0 0 16 0 0 0 0 16
,
 16 15 0 0 0 1 0 0 0 0 16 0 0 0 0 16
`G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,13,9,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[2,0,0,0,0,2,0,0,0,0,1,0,0,0,16,16],[13,4,0,0,9,4,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,15,1,0,0,0,0,16,0,0,0,0,16] >;`

C42.674C23 in GAP, Magma, Sage, TeX

`C_4^2._{674}C_2^3`
`% in TeX`

`G:=Group("C4^2.674C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1638);`
`// by ID`

`G=gap.SmallGroup(128,1638);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,521,124]);`
`// 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,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=b^2*d>;`
`// generators/relations`

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