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## G = C42.307C23order 128 = 27

### 168th 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 — C42.307C23
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C2×C42 — C23.36C23 — C42.307C23
 Lower central C1 — C22 — C42.307C23
 Upper central C1 — C2×C4 — C42.307C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.307C23

Generators and relations for C42.307C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, ac=ca, dad=a-1b2, eae=ab2, bc=cb, bd=db, be=eb, cd=dc, ece=a2b2c, ede=a2b2d >

Subgroups: 252 in 178 conjugacy classes, 126 normal (40 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×8], C8 [×8], C2×C4 [×6], C2×C4 [×6], C2×C4 [×9], D4 [×5], Q8, C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×8], C2×C8 [×4], M4(2) [×2], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4×C8 [×2], C8⋊C4 [×4], C22⋊C8 [×6], C4⋊C8 [×2], C4⋊C8 [×8], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2 [×2], C22×C8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×C4⋊C8, C42.6C22 [×2], C42.6C4, C42.7C22 [×2], C8×D4 [×2], C89D4 [×4], C84Q8 [×2], C23.36C23, C42.307C23
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- 1+4, C23.33C23, C2×C8○D4, Q8○M4(2), C42.307C23

Smallest permutation representation of C42.307C23
On 64 points
Generators in S64
```(1 14 51 57)(2 15 52 58)(3 16 53 59)(4 9 54 60)(5 10 55 61)(6 11 56 62)(7 12 49 63)(8 13 50 64)(17 47 27 33)(18 48 28 34)(19 41 29 35)(20 42 30 36)(21 43 31 37)(22 44 32 38)(23 45 25 39)(24 46 26 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 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(1 43)(2 34)(3 45)(4 36)(5 47)(6 38)(7 41)(8 40)(9 24)(10 31)(11 18)(12 25)(13 20)(14 27)(15 22)(16 29)(17 57)(19 59)(21 61)(23 63)(26 60)(28 62)(30 64)(32 58)(33 55)(35 49)(37 51)(39 53)(42 54)(44 56)(46 50)(48 52)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 4 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 C4 C4 C8○D4 2+ 1+4 2- 1+4 Q8○M4(2) kernel C42.307C23 C2×C4⋊C8 C42.6C22 C42.6C4 C42.7C22 C8×D4 C8⋊9D4 C8⋊4Q8 C23.36C23 C4⋊D4 C22⋊Q8 C22.D4 C4.4D4 C42.C2 C42⋊2C2 C22 C4 C4 C2 # reps 1 1 2 1 2 2 4 2 1 2 2 4 2 2 4 8 1 1 2

Matrix representation of C42.307C23 in GL6(𝔽17)

 13 0 0 0 0 0 0 4 0 0 0 0 0 0 4 1 0 0 0 0 0 13 0 0 0 0 9 8 1 13 0 0 7 9 9 16
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 1 0 0 0 9 8 14 13 0 0 16 0 0 0 0 0 6 12 9 9
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 4 1 0 0 0 0 2 13 0 0 0 0 9 8 1 13 0 0 11 9 0 16
,
 0 4 0 0 0 0 13 0 0 0 0 0 0 0 16 4 0 0 0 0 0 1 0 0 0 0 15 2 13 16 0 0 10 3 15 4

`G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,9,7,0,0,1,13,8,9,0,0,0,0,1,9,0,0,0,0,13,16],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,16,6,0,0,0,8,0,12,0,0,1,14,0,9,0,0,0,13,0,9],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,2,9,11,0,0,1,13,8,9,0,0,0,0,1,0,0,0,0,0,13,16],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,16,0,15,10,0,0,4,1,2,3,0,0,0,0,13,15,0,0,0,0,16,4] >;`

C42.307C23 in GAP, Magma, Sage, TeX

`C_4^2._{307}C_2^3`
`% in TeX`

`G:=Group("C4^2.307C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1724);`
`// by ID`

`G=gap.SmallGroup(128,1724);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,219,675,1018,80,124]);`
`// Polycyclic`

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

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