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## G = C42.12C23order 128 = 27

### 12nd non-split extension by C42 of C23 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.12C23
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2.C25 — C42.12C23
 Lower central C1 — C2 — C2×C4 — C42.12C23
 Upper central C1 — C2 — C22×C4 — C42.12C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.12C23

Generators and relations for C42.12C23
G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=b2, cac=ab=ba, dad=a-1, eae-1=ab2, cbc=dbd=b-1, be=eb, dcd=b-1c, ce=ec, ede-1=b2d >

Subgroups: 780 in 368 conjugacy classes, 106 normal (18 characteristic)
C1, C2, C2 [×11], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×25], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×28], D4 [×4], D4 [×38], Q8 [×4], Q8 [×8], C23, C23 [×2], C23 [×12], C42 [×2], C22⋊C4 [×5], C4⋊C4, C2×C8 [×2], M4(2) [×4], M4(2) [×2], D8 [×8], SD16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×6], C2×D4 [×29], C2×Q8, C2×Q8 [×2], C2×Q8 [×6], C4○D4 [×8], C4○D4 [×36], C24, C4.D4 [×2], C4.10D4 [×2], C4≀C2 [×8], C42⋊C2, C22≀C2 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4 [×2], C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C8⋊C22 [×4], C22×D4, C2×C4○D4, C2×C4○D4 [×2], C2×C4○D4 [×6], 2+ 1+4 [×2], 2+ 1+4 [×4], 2- 1+4 [×2], 2- 1+4 [×2], M4(2).8C22, C42⋊C22 [×2], D44D4 [×4], D4.8D4 [×4], C22.29C24, C2×C8⋊C22 [×2], C2.C25, C42.12C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, C42.12C23

Character table of C42.12C23

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D size 1 1 2 2 2 4 4 4 4 4 4 8 8 2 2 2 2 4 4 4 4 4 4 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 1 -1 1 -1 -1 1 1 -1 linear of order 2 ρ5 1 1 1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ6 1 1 -1 1 -1 1 1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 linear of order 2 ρ7 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ8 1 1 -1 1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ9 1 1 -1 1 -1 -1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 1 1 -1 1 1 -1 1 -1 -1 1 linear of order 2 ρ10 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ11 1 1 -1 1 -1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 1 1 -1 linear of order 2 ρ12 1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ13 1 1 -1 1 -1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 linear of order 2 ρ14 1 1 1 1 1 -1 -1 1 -1 1 -1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ15 1 1 -1 1 -1 1 -1 -1 1 1 1 -1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 linear of order 2 ρ16 1 1 1 1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ17 2 2 -2 -2 2 0 2 0 0 0 2 0 0 -2 -2 2 2 0 0 0 0 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 2 0 -2 0 0 0 -2 0 0 -2 -2 2 2 0 0 0 0 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 -2 -2 0 0 -2 0 -2 0 0 0 -2 2 -2 2 2 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 -2 2 0 0 -2 0 2 0 0 0 2 2 -2 -2 -2 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 -2 2 -2 -2 0 0 2 0 0 0 0 2 -2 -2 2 0 -2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 2 2 2 -2 0 0 2 0 0 0 0 -2 -2 -2 -2 0 2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 2 -2 -2 0 2 0 0 0 -2 0 0 2 -2 2 -2 0 0 0 0 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 2 2 2 2 0 0 -2 0 0 0 0 -2 -2 -2 -2 0 -2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 -2 2 -2 2 0 0 -2 0 0 0 0 2 -2 -2 2 0 2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ26 2 2 2 -2 -2 0 -2 0 0 0 2 0 0 2 -2 2 -2 0 0 0 0 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ27 2 2 -2 -2 2 0 0 2 0 -2 0 0 0 2 2 -2 -2 2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ28 2 2 2 -2 -2 0 0 2 0 2 0 0 0 -2 2 -2 2 -2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C42.12C23
On 16 points - transitive group 16T269
Generators in S16
```(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 4 2 3)(5 8 6 7)(9 12 11 10)(13 14 15 16)
(1 12)(2 10)(3 11)(4 9)(5 13)(6 15)(7 14)(8 16)
(3 4)(5 6)(9 10)(11 12)(13 16)(14 15)
(1 6 2 5)(3 8 4 7)(9 14 11 16)(10 13 12 15)```

`G:=sub<Sym(16)| (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,2,3)(5,8,6,7)(9,12,11,10)(13,14,15,16), (1,12)(2,10)(3,11)(4,9)(5,13)(6,15)(7,14)(8,16), (3,4)(5,6)(9,10)(11,12)(13,16)(14,15), (1,6,2,5)(3,8,4,7)(9,14,11,16)(10,13,12,15)>;`

`G:=Group( (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,2,3)(5,8,6,7)(9,12,11,10)(13,14,15,16), (1,12)(2,10)(3,11)(4,9)(5,13)(6,15)(7,14)(8,16), (3,4)(5,6)(9,10)(11,12)(13,16)(14,15), (1,6,2,5)(3,8,4,7)(9,14,11,16)(10,13,12,15) );`

`G=PermutationGroup([(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,4,2,3),(5,8,6,7),(9,12,11,10),(13,14,15,16)], [(1,12),(2,10),(3,11),(4,9),(5,13),(6,15),(7,14),(8,16)], [(3,4),(5,6),(9,10),(11,12),(13,16),(14,15)], [(1,6,2,5),(3,8,4,7),(9,14,11,16),(10,13,12,15)])`

`G:=TransitiveGroup(16,269);`

On 16 points - transitive group 16T282
Generators in S16
```(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 5 8 3)(2 6 7 4)(9 14 11 16)(10 15 12 13)
(1 12)(2 14)(3 13)(4 11)(5 15)(6 9)(7 16)(8 10)
(1 8)(2 7)(9 14)(10 13)(11 16)(12 15)
(1 5 8 3)(2 4 7 6)(9 14 11 16)(10 13 12 15)```

`G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,5,8,3)(2,6,7,4)(9,14,11,16)(10,15,12,13), (1,12)(2,14)(3,13)(4,11)(5,15)(6,9)(7,16)(8,10), (1,8)(2,7)(9,14)(10,13)(11,16)(12,15), (1,5,8,3)(2,4,7,6)(9,14,11,16)(10,13,12,15)>;`

`G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,5,8,3)(2,6,7,4)(9,14,11,16)(10,15,12,13), (1,12)(2,14)(3,13)(4,11)(5,15)(6,9)(7,16)(8,10), (1,8)(2,7)(9,14)(10,13)(11,16)(12,15), (1,5,8,3)(2,4,7,6)(9,14,11,16)(10,13,12,15) );`

`G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,5,8,3),(2,6,7,4),(9,14,11,16),(10,15,12,13)], [(1,12),(2,14),(3,13),(4,11),(5,15),(6,9),(7,16),(8,10)], [(1,8),(2,7),(9,14),(10,13),(11,16),(12,15)], [(1,5,8,3),(2,4,7,6),(9,14,11,16),(10,13,12,15)])`

`G:=TransitiveGroup(16,282);`

Matrix representation of C42.12C23 in GL8(ℤ)

 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
,
 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 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 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0

`G:=sub<GL(8,Integers())| [0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0] >;`

C42.12C23 in GAP, Magma, Sage, TeX

`C_4^2._{12}C_2^3`
`% in TeX`

`G:=Group("C4^2.12C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1753);`
`// by ID`

`G=gap.SmallGroup(128,1753);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2019,248,2804,1411,718,172,2028]);`
`// Polycyclic`

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

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