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## G = C42.308C23order 128 = 27

### 169th 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 — C42.308C23
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C2×C42 — C23.36C23 — C42.308C23
 Lower central C1 — C22 — C42.308C23
 Upper central C1 — C2×C4 — C42.308C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.308C23

Generators and relations for C42.308C23
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, dcd=b2c, ece=a2b2c, ede=a2b2d >

Subgroups: 252 in 177 conjugacy classes, 126 normal (52 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×9], C8 [×8], C2×C4 [×6], C2×C4 [×6], C2×C4 [×9], D4 [×4], Q8 [×2], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], C2×C8 [×2], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4×C8 [×2], C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×2], C22⋊C8 [×4], C4⋊C8 [×4], C4⋊C8 [×6], 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], C2×M4(2) [×4], C4⋊M4(2), C42.6C22 [×2], C42.12C4, C42.7C22 [×2], C8×D4, C89D4 [×2], C86D4, C86D4 [×2], C8×Q8, C84Q8, C23.36C23, C42.308C23
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), C42.308C23

Smallest permutation representation of C42.308C23
On 64 points
Generators in S64
```(1 39 51 45)(2 40 52 46)(3 33 53 47)(4 34 54 48)(5 35 55 41)(6 36 56 42)(7 37 49 43)(8 38 50 44)(9 32 60 22)(10 25 61 23)(11 26 62 24)(12 27 63 17)(13 28 64 18)(14 29 57 19)(15 30 58 20)(16 31 59 21)
(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)
(2 6)(4 8)(9 60)(10 57)(11 62)(12 59)(13 64)(14 61)(15 58)(16 63)(18 22)(20 24)(26 30)(28 32)(33 43)(34 48)(35 45)(36 42)(37 47)(38 44)(39 41)(40 46)(50 54)(52 56)
(1 16)(2 64)(3 10)(4 58)(5 12)(6 60)(7 14)(8 62)(9 56)(11 50)(13 52)(15 54)(17 45)(18 36)(19 47)(20 38)(21 41)(22 40)(23 43)(24 34)(25 37)(26 48)(27 39)(28 42)(29 33)(30 44)(31 35)(32 46)(49 57)(51 59)(53 61)(55 63)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4Q 8A ··· 8H 8I ··· 8T order 1 2 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 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 1 1 1 1 1 2 4 4 4 type + + + + + + + + + + + + - image C1 C2 C2 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 C42.308C23 C4⋊M4(2) C42.6C22 C42.12C4 C42.7C22 C8×D4 C8⋊9D4 C8⋊6D4 C8×Q8 C8⋊4Q8 C23.36C23 C4⋊D4 C22⋊Q8 C22.D4 C4.4D4 C42.C2 C42⋊2C2 C4 C4 C4 C2 # reps 1 1 2 1 2 1 2 3 1 1 1 2 2 4 2 2 4 8 1 1 2

Matrix representation of C42.308C23 in GL6(𝔽17)

 0 2 0 0 0 0 9 0 0 0 0 0 0 0 0 4 0 0 0 0 4 0 0 0 0 0 15 10 4 13 0 0 8 8 0 13
,
 4 0 0 0 0 0 0 4 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
,
 0 4 0 0 0 0 1 0 0 0 0 0 0 0 7 13 15 0 0 0 14 2 15 2 0 0 14 16 14 13 0 0 2 16 13 11
,
 1 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 4 1 0 0 0 3 4 2 16
,
 0 8 0 0 0 0 15 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 5 13 16 1 0 0 9 8 0 1

`G:=sub<GL(6,GF(17))| [0,9,0,0,0,0,2,0,0,0,0,0,0,0,0,4,15,8,0,0,4,0,10,8,0,0,0,0,4,0,0,0,0,0,13,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,7,14,14,2,0,0,13,2,16,16,0,0,15,15,14,13,0,0,0,2,13,11],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,3,0,0,0,16,4,4,0,0,0,0,1,2,0,0,0,0,0,16],[0,15,0,0,0,0,8,0,0,0,0,0,0,0,0,1,5,9,0,0,1,0,13,8,0,0,0,0,16,0,0,0,0,0,1,1] >;`

C42.308C23 in GAP, Magma, Sage, TeX

`C_4^2._{308}C_2^3`
`% in TeX`

`G:=Group("C4^2.308C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1725);`
`// by ID`

`G=gap.SmallGroup(128,1725);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,219,675,1018,80,124]);`
`// 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,d*c*d=b^2*c,e*c*e=a^2*b^2*c,e*d*e=a^2*b^2*d>;`
`// generators/relations`

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