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## G = C42.301C23order 128 = 27

### 162nd 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 — C42.301C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C22.26C24 — C42.301C23
 Lower central C1 — C22 — C42.301C23
 Upper central C1 — C2×C4 — C42.301C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.301C23

Generators and relations for C42.301C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1b2, dad=a-1, ae=ea, bc=cb, bd=db, be=eb, dcd=ece=a2b2c, ede=b2d >

Subgroups: 332 in 201 conjugacy classes, 126 normal (28 characteristic)
C1, C2 [×3], C2 [×5], C4 [×4], C4 [×8], C22, C22 [×15], C8 [×8], C2×C4 [×6], C2×C4 [×4], C2×C4 [×13], D4 [×12], Q8 [×2], C23, C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×4], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×12], C4⋊C8 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C22×C8 [×4], C2×M4(2) [×4], C2×C4○D4 [×2], (C22×C8)⋊C2 [×4], C42.12C4, C42.6C4, C8×D4 [×2], C89D4 [×4], C86D4 [×2], C22.26C24, C42.301C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C8○D4 [×2], C23×C4, 2+ 1+4 [×2], C22.11C24, C2×C8○D4, Q8○M4(2), C42.301C23

Smallest permutation representation of C42.301C23
On 64 points
Generators in S64
```(1 12 51 37)(2 34 52 9)(3 14 53 39)(4 36 54 11)(5 16 55 33)(6 38 56 13)(7 10 49 35)(8 40 50 15)(17 43 29 63)(18 60 30 48)(19 45 31 57)(20 62 32 42)(21 47 25 59)(22 64 26 44)(23 41 27 61)(24 58 28 46)
(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 21)(2 30)(3 23)(4 32)(5 17)(6 26)(7 19)(8 28)(9 48)(10 57)(11 42)(12 59)(13 44)(14 61)(15 46)(16 63)(18 52)(20 54)(22 56)(24 50)(25 51)(27 53)(29 55)(31 49)(33 43)(34 60)(35 45)(36 62)(37 47)(38 64)(39 41)(40 58)
(1 23)(2 32)(3 17)(4 26)(5 19)(6 28)(7 21)(8 30)(9 62)(10 47)(11 64)(12 41)(13 58)(14 43)(15 60)(16 45)(18 50)(20 52)(22 54)(24 56)(25 49)(27 51)(29 53)(31 55)(33 57)(34 42)(35 59)(36 44)(37 61)(38 46)(39 63)(40 48)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 C8○D4 2+ 1+4 Q8○M4(2) kernel C42.301C23 (C22×C8)⋊C2 C42.12C4 C42.6C4 C8×D4 C8⋊9D4 C8⋊6D4 C22.26C24 C4⋊D4 C4.4D4 C4⋊1D4 C4⋊Q8 C4 C4 C2 # reps 1 4 1 1 2 4 2 1 8 4 2 2 8 2 2

Matrix representation of C42.301C23 in GL6(𝔽17)

 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1 0 0 0
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 0 0 0 0 0 0 2 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1
,
 0 4 0 0 0 0 13 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 16 0
,
 0 1 0 0 0 0 1 0 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))| [0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C42.301C23 in GAP, Magma, Sage, TeX

`C_4^2._{301}C_2^3`
`% in TeX`

`G:=Group("C4^2.301C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1713);`
`// by ID`

`G=gap.SmallGroup(128,1713);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,891,675,1018,521,124]);`
`// 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*b^2,d*a*d=a^-1,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=e*c*e=a^2*b^2*c,e*d*e=b^2*d>;`
`// generators/relations`

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