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## G = C42.678C23order 128 = 27

### 93rd 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 — C42.678C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C42⋊C2 — C4×C4○D4 — C42.678C23
 Lower central C1 — C22 — C42.678C23
 Upper central C1 — C2×C4 — C42.678C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.678C23

Generators and relations for C42.678C23
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, dcd=a2c, ce=ec, ede=b2d >

Subgroups: 268 in 196 conjugacy classes, 134 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×8], C8 [×8], C2×C4 [×6], C2×C4 [×10], C2×C4 [×13], D4 [×6], Q8 [×2], C23, C23 [×2], C42 [×4], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×6], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C4×C8 [×4], C8⋊C4 [×4], C22⋊C8 [×8], C4⋊C8 [×2], C4⋊C8 [×6], C2×C42, C2×C42 [×2], C42⋊C2, C42⋊C2 [×2], C4×D4 [×2], C4×D4 [×4], C4×Q8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×C4○D4, C82M4(2) [×2], (C22×C8)⋊C2 [×2], C2×C4⋊C8, C4⋊M4(2), C42.12C4 [×2], C42.6C4 [×2], C42.7C22 [×4], C4×C4○D4, C42.678C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C8○D4 [×2], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, C2×C8○D4, Q8○M4(2), C42.678C23

Smallest permutation representation of C42.678C23
On 64 points
Generators in S64
```(1 63 55 10)(2 11 56 64)(3 57 49 12)(4 13 50 58)(5 59 51 14)(6 15 52 60)(7 61 53 16)(8 9 54 62)(17 39 31 42)(18 43 32 40)(19 33 25 44)(20 45 26 34)(21 35 27 46)(22 47 28 36)(23 37 29 48)(24 41 30 38)
(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 56)(4 50)(6 52)(8 54)(9 62)(11 64)(13 58)(15 60)(17 21)(18 28)(19 23)(20 30)(22 32)(24 26)(25 29)(27 31)(33 37)(34 41)(35 39)(36 43)(38 45)(40 47)(42 46)(44 48)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4R 4S 4T 4U 4V 8A ··· 8H 8I ··· 8T order 1 2 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 2 2 4 4 1 1 1 1 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4○D4 C8○D4 Q8○M4(2) kernel C42.678C23 C8○2M4(2) (C22×C8)⋊C2 C2×C4⋊C8 C4⋊M4(2) C42.12C4 C42.6C4 C42.7C22 C4×C4○D4 C4×D4 C4×Q8 C2×C4 C4 C2 # reps 1 2 2 1 1 2 2 4 1 12 4 8 8 2

Matrix representation of C42.678C23 in GL4(𝔽17) generated by

 13 0 0 0 13 4 0 0 0 0 1 0 0 0 0 1
,
 16 0 0 0 0 16 0 0 0 0 13 0 0 0 0 13
,
 1 15 0 0 1 16 0 0 0 0 8 0 0 0 0 8
,
 1 0 0 0 1 16 0 0 0 0 1 0 0 0 0 16
,
 16 0 0 0 0 16 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(17))| [13,13,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[1,1,0,0,15,16,0,0,0,0,8,0,0,0,0,8],[1,1,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,0,1,0,0,1,0] >;`

C42.678C23 in GAP, Magma, Sage, TeX

`C_4^2._{678}C_2^3`
`% in TeX`

`G:=Group("C4^2.678C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1657);`
`// by ID`

`G=gap.SmallGroup(128,1657);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,1018,521,124]);`
`// 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,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=a^2*c,c*e=e*c,e*d*e=b^2*d>;`
`// generators/relations`

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