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## G = C42.354C23order 128 = 27

### 215th non-split extension by C42 of C23 acting via C23/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.354C23
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — C23.33C23 — C42.354C23
 Lower central C1 — C2 — C2×C4 — C42.354C23
 Upper central C1 — C22 — C42⋊C2 — C42.354C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.354C23

Generators and relations for C42.354C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=b2, ab=ba, ac=ca, dad-1=ab2, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=a2c, ece=bc, de=ed >

Subgroups: 324 in 186 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×13], C22, C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×2], D4 [×5], Q8 [×2], Q8 [×5], C23, C23, C42 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×11], C2×C8 [×4], SD16 [×4], Q16 [×4], C22×C4, C22×C4 [×4], C2×D4, C2×D4, C2×Q8 [×3], C4○D4 [×4], C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×4], C22⋊Q8 [×2], C42.C2, C42.C2 [×2], C422C2 [×2], C4⋊Q8, C2×SD16 [×2], C2×Q16 [×2], C2×C4○D4, C42.7C22, C4×SD16, C4×Q16, SD16⋊C4, Q16⋊C4, D4.7D4 [×2], D4.D4, C42Q16, D4.Q8, Q8.Q8, C23.47D4, C23.48D4, C23.33C23, C22.35C24, C42.354C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.19C24, D4○SD16, Q8○D8, C42.354C23

Smallest permutation representation of C42.354C23
On 64 points
Generators in S64
```(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 22 25 20)(2 23 26 17)(3 24 27 18)(4 21 28 19)(5 15 63 10)(6 16 64 11)(7 13 61 12)(8 14 62 9)(29 40 42 35)(30 37 43 36)(31 38 44 33)(32 39 41 34)(45 52 57 54)(46 49 58 55)(47 50 59 56)(48 51 60 53)
(1 57 25 45)(2 58 26 46)(3 59 27 47)(4 60 28 48)(5 44 63 31)(6 41 64 32)(7 42 61 29)(8 43 62 30)(9 36 14 37)(10 33 15 38)(11 34 16 39)(12 35 13 40)(17 55 23 49)(18 56 24 50)(19 53 21 51)(20 54 22 52)
(1 32 25 41)(2 42 26 29)(3 30 27 43)(4 44 28 31)(5 58 63 46)(6 47 64 59)(7 60 61 48)(8 45 62 57)(9 54 14 52)(10 49 15 55)(11 56 16 50)(12 51 13 53)(17 40 23 35)(18 36 24 37)(19 38 21 33)(20 34 22 39)
(5 10)(6 11)(7 12)(8 9)(13 61)(14 62)(15 63)(16 64)(17 23)(18 24)(19 21)(20 22)(33 38)(34 39)(35 40)(36 37)(45 54)(46 55)(47 56)(48 53)(49 58)(50 59)(51 60)(52 57)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A ··· 4F 4G ··· 4O 4P 4Q 4R 4S 8A 8B 8C 8D 8E 8F order 1 2 2 2 2 2 2 4 ··· 4 4 ··· 4 4 4 4 4 8 8 8 8 8 8 size 1 1 1 1 4 4 4 2 ··· 2 4 ··· 4 8 8 8 8 4 4 4 4 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 C4○D4 D4○SD16 Q8○D8 kernel C42.354C23 C42.7C22 C4×SD16 C4×Q16 SD16⋊C4 Q16⋊C4 D4.7D4 D4.D4 C4⋊2Q16 D4.Q8 Q8.Q8 C23.47D4 C23.48D4 C23.33C23 C22.35C24 C22⋊C4 C4⋊C4 D4 Q8 C2 C2 # reps 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 2 4 4 2 2

Matrix representation of C42.354C23 in GL6(𝔽17)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 8 10 0 0 0 0 9 9 0 0 0 0 0 0 5 12 0 0 0 0 12 12 0 0 0 0 0 0 5 12 0 0 0 0 12 12
,
 1 2 0 0 0 0 0 16 0 0 0 0 0 0 16 0 6 0 0 0 0 16 0 6 0 0 11 0 1 0 0 0 0 11 0 1
,
 16 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 0 0 0 0 0 16

`G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[8,9,0,0,0,0,10,9,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,0,0,0,0,5,12,0,0,0,0,12,12],[1,0,0,0,0,0,2,16,0,0,0,0,0,0,16,0,11,0,0,0,0,16,0,11,0,0,6,0,1,0,0,0,0,6,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;`

C42.354C23 in GAP, Magma, Sage, TeX

`C_4^2._{354}C_2^3`
`% in TeX`

`G:=Group("C4^2.354C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1852);`
`// by ID`

`G=gap.SmallGroup(128,1852);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,1018,80,4037,1027,124]);`
`// Polycyclic`

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

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