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## G = C42.18C23order 128 = 27

### 18th non-split extension by C42 of C23 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.18C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C23.33C23 — C42.18C23
 Lower central C1 — C2 — C2×C4 — C42.18C23
 Upper central C1 — C22 — C42⋊C2 — C42.18C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.18C23

Generators and relations for C42.18C23
G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=a2, ab=ba, cac=a-1, ad=da, eae-1=ab2, cbc=dbd=b-1, be=eb, dcd=bc, ece-1=a2c, de=ed >

Subgroups: 548 in 252 conjugacy classes, 100 normal (38 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×19], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×13], D4 [×2], D4 [×19], Q8 [×2], Q8 [×3], C23, C23 [×8], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], D8 [×6], SD16 [×8], Q16 [×2], C22×C4, C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×D4 [×11], C2×Q8 [×2], C4○D4 [×4], C4○D4 [×6], C24, D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×4], C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C22≀C2 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4 [×2], C22×C8, C2×M4(2), C2×D8, C2×D8 [×2], C2×SD16 [×2], C2×SD16 [×2], C2×Q16, C4○D8 [×4], C8⋊C22 [×4], C22×D4, C2×C4○D4 [×2], C2×D4⋊C4, C23.36D4, C42.6C22, C4⋊D8 [×2], C4⋊SD16 [×2], D4.2D4 [×2], Q8.D4 [×2], C23.33C23, C22.29C24, C2×C4○D8, C2×C8⋊C22, C42.18C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D4○D8, D4○SD16, C42.18C23

Smallest permutation representation of C42.18C23
On 32 points
Generators in S32
```(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)
(1 12 16 27)(2 9 13 28)(3 10 14 25)(4 11 15 26)(5 20 22 32)(6 17 23 29)(7 18 24 30)(8 19 21 31)
(1 28)(2 27)(3 26)(4 25)(5 8)(6 7)(9 16)(10 15)(11 14)(12 13)(17 30)(18 29)(19 32)(20 31)(21 22)(23 24)
(1 17)(2 18)(3 19)(4 20)(5 11)(6 12)(7 9)(8 10)(13 30)(14 31)(15 32)(16 29)(21 25)(22 26)(23 27)(24 28)
(1 26 3 28)(2 12 4 10)(5 31 7 29)(6 20 8 18)(9 16 11 14)(13 27 15 25)(17 22 19 24)(21 30 23 32)```

`G:=sub<Sym(32)| (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), (1,12,16,27)(2,9,13,28)(3,10,14,25)(4,11,15,26)(5,20,22,32)(6,17,23,29)(7,18,24,30)(8,19,21,31), (1,28)(2,27)(3,26)(4,25)(5,8)(6,7)(9,16)(10,15)(11,14)(12,13)(17,30)(18,29)(19,32)(20,31)(21,22)(23,24), (1,17)(2,18)(3,19)(4,20)(5,11)(6,12)(7,9)(8,10)(13,30)(14,31)(15,32)(16,29)(21,25)(22,26)(23,27)(24,28), (1,26,3,28)(2,12,4,10)(5,31,7,29)(6,20,8,18)(9,16,11,14)(13,27,15,25)(17,22,19,24)(21,30,23,32)>;`

`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), (1,12,16,27)(2,9,13,28)(3,10,14,25)(4,11,15,26)(5,20,22,32)(6,17,23,29)(7,18,24,30)(8,19,21,31), (1,28)(2,27)(3,26)(4,25)(5,8)(6,7)(9,16)(10,15)(11,14)(12,13)(17,30)(18,29)(19,32)(20,31)(21,22)(23,24), (1,17)(2,18)(3,19)(4,20)(5,11)(6,12)(7,9)(8,10)(13,30)(14,31)(15,32)(16,29)(21,25)(22,26)(23,27)(24,28), (1,26,3,28)(2,12,4,10)(5,31,7,29)(6,20,8,18)(9,16,11,14)(13,27,15,25)(17,22,19,24)(21,30,23,32) );`

`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)], [(1,12,16,27),(2,9,13,28),(3,10,14,25),(4,11,15,26),(5,20,22,32),(6,17,23,29),(7,18,24,30),(8,19,21,31)], [(1,28),(2,27),(3,26),(4,25),(5,8),(6,7),(9,16),(10,15),(11,14),(12,13),(17,30),(18,29),(19,32),(20,31),(21,22),(23,24)], [(1,17),(2,18),(3,19),(4,20),(5,11),(6,12),(7,9),(8,10),(13,30),(14,31),(15,32),(16,29),(21,25),(22,26),(23,27),(24,28)], [(1,26,3,28),(2,12,4,10),(5,31,7,29),(6,20,8,18),(9,16,11,14),(13,27,15,25),(17,22,19,24),(21,30,23,32)])`

32 conjugacy classes

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

32 irreducible representations

 dim 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 D4 D4 D4 C4○D4 D4○D8 D4○SD16 kernel C42.18C23 C2×D4⋊C4 C23.36D4 C42.6C22 C4⋊D8 C4⋊SD16 D4.2D4 Q8.D4 C23.33C23 C22.29C24 C2×C4○D8 C2×C8⋊C22 C22⋊C4 C4⋊C4 C4○D4 C2×C4 C2 C2 # reps 1 1 1 1 2 2 2 2 1 1 1 1 2 2 4 4 2 2

Matrix representation of C42.18C23 in GL6(𝔽17)

 13 8 0 0 0 0 0 4 0 0 0 0 0 0 10 0 10 10 0 0 5 0 0 10 0 0 0 12 12 12 0 0 12 5 12 12
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 15 0 0 0 0 1 16 0 0 0 0 16 1 0 16 0 0 0 1 1 0
,
 4 9 0 0 0 0 4 13 0 0 0 0 0 0 0 0 7 10 0 0 5 0 0 10 0 0 5 12 5 12 0 0 0 12 5 12
,
 1 15 0 0 0 0 0 16 0 0 0 0 0 0 16 0 15 0 0 0 0 0 16 1 0 0 0 0 1 0 0 0 0 1 1 0
,
 4 9 0 0 0 0 0 13 0 0 0 0 0 0 0 0 7 10 0 0 12 0 7 0 0 0 0 12 5 12 0 0 5 12 5 12

`G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,8,4,0,0,0,0,0,0,10,5,0,12,0,0,0,0,12,5,0,0,10,0,12,12,0,0,10,10,12,12],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,16,0,0,0,15,16,1,1,0,0,0,0,0,1,0,0,0,0,16,0],[4,4,0,0,0,0,9,13,0,0,0,0,0,0,0,5,5,0,0,0,0,0,12,12,0,0,7,0,5,5,0,0,10,10,12,12],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,16,1,1,0,0,0,1,0,0],[4,0,0,0,0,0,9,13,0,0,0,0,0,0,0,12,0,5,0,0,0,0,12,12,0,0,7,7,5,5,0,0,10,0,12,12] >;`

C42.18C23 in GAP, Magma, Sage, TeX

`C_4^2._{18}C_2^3`
`% in TeX`

`G:=Group("C4^2.18C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1777);`
`// by ID`

`G=gap.SmallGroup(128,1777);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,1018,248,4037,1027,124]);`
`// Polycyclic`

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

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