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G = C23.19D4order 64 = 26

12nd non-split extension by C23 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C23.19D4
 Chief series C1 — C2 — C4 — C2×C4 — C4⋊C4 — C42⋊C2 — C23.19D4
 Lower central C1 — C2 — C2×C4 — C23.19D4
 Upper central C1 — C22 — C22×C4 — C23.19D4
 Jennings C1 — C2 — C2 — C2×C4 — C23.19D4

Generators and relations for C23.19D4
G = < a,b,c,d,e | a2=b2=c2=e2=1, d4=c, dad-1=ab=ba, ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=bcd3 >

Character table of C23.19D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D size 1 1 1 1 4 8 2 2 2 2 4 4 4 4 8 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 2 2 2 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 0 -2 2 0 0 0 2i -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ12 2 -2 2 -2 0 0 -2 2 0 0 0 -2i 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ13 2 -2 2 -2 0 0 2 -2 0 0 2i 0 0 -2i 0 0 0 0 0 complex lifted from C4○D4 ρ14 2 -2 2 -2 0 0 2 -2 0 0 -2i 0 0 2i 0 0 0 0 0 complex lifted from C4○D4 ρ15 2 -2 -2 2 0 0 0 0 2i -2i 0 0 0 0 0 -√2 √2 √-2 -√-2 complex lifted from C4○D8 ρ16 2 -2 -2 2 0 0 0 0 -2i 2i 0 0 0 0 0 -√2 √2 -√-2 √-2 complex lifted from C4○D8 ρ17 2 -2 -2 2 0 0 0 0 2i -2i 0 0 0 0 0 √2 -√2 -√-2 √-2 complex lifted from C4○D8 ρ18 2 -2 -2 2 0 0 0 0 -2i 2i 0 0 0 0 0 √2 -√2 √-2 -√-2 complex lifted from C4○D8 ρ19 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22

Smallest permutation representation of C23.19D4
On 32 points
Generators in S32
```(1 30)(2 13)(3 32)(4 15)(5 26)(6 9)(7 28)(8 11)(10 21)(12 23)(14 17)(16 19)(18 25)(20 27)(22 29)(24 31)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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)
(2 22)(3 7)(4 20)(6 18)(8 24)(9 11)(10 28)(12 26)(13 15)(14 32)(16 30)(17 21)(25 31)(27 29)```

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

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

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

C23.19D4 is a maximal subgroup of
C24.115D4  C24.116D4  C24.117D4  C42.229D4  C42.233D4  C24.121D4  C24.124D4  C24.127D4  C24.130D4  C42.287D4  C42.292D4
(C2×D4).D2p: (C2×D4).301D4  (C2×D4).303D4  (C2×D4).304D4  C4.2+ 1+4  C4.142+ 1+4  C4.152+ 1+4  C4.162+ 1+4  C42.461C23 ...
C4⋊C4.D2p: C42.352C23  C42.355C23  C42.356C23  C42.360C23  C42.423C23  C42.425C23  C42.426C23  C4.Q8⋊S3 ...
C2p.(C4○D8): C42.384D4  C42.450D4  C42.280D4  C42.285D4  C23.18D12  C23.13D20  C23.13D28 ...
C23.19D4 is a maximal quotient of
C24.69D4  C24.74D4  C4.Q810C4  C2.D85C4  C428C4⋊C2  C4⋊C4.Q8
C4⋊C4.D2p: C24.71D4  D4⋊C4⋊C4  C4.67(C4×D4)  C24.83D4  C24.84D4  (C2×C8).24Q8  D6⋊C811C2  C241C4⋊C2 ...
(C2×C8).D2p: C24.88D4  C24.89D4  (C2×C8).168D4  C23.18D12  C23.13D20  C23.13D28 ...

Matrix representation of C23.19D4 in GL4(𝔽17) generated by

 1 15 0 0 0 16 0 0 0 0 13 8 0 0 13 4
,
 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16
,
 13 0 0 0 13 4 0 0 0 0 0 11 0 0 3 11
,
 1 0 0 0 1 16 0 0 0 0 1 0 0 0 1 16
`G:=sub<GL(4,GF(17))| [1,0,0,0,15,16,0,0,0,0,13,13,0,0,8,4],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[13,13,0,0,0,4,0,0,0,0,0,3,0,0,11,11],[1,1,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;`

C23.19D4 in GAP, Magma, Sage, TeX

`C_2^3._{19}D_4`
`% in TeX`

`G:=Group("C2^3.19D4");`
`// GroupNames label`

`G:=SmallGroup(64,163);`
`// by ID`

`G=gap.SmallGroup(64,163);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,121,199,362,50,1444,376,88]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=e^2=1,d^4=c,d*a*d^-1=a*b=b*a,a*c=c*a,e*a*e=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b*c*d^3>;`
`// generators/relations`

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