Copied to
clipboard

## G = C23.D18order 288 = 25·32

### 5th non-split extension by C23 of D18 acting via D18/C6=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×C3.A4 — C23.D18
 Chief series C1 — C22 — C2×C6 — C3.A4 — C2×C3.A4 — C2×C3.S4 — C23.D18
 Lower central C3.A4 — C2×C3.A4 — C23.D18
 Upper central C1 — C2 — C22

Generators and relations for C23.D18
G = < a,b,c,d,e | a2=b2=c2=d18=1, e2=a, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ece-1=bc=cb, be=eb, dcd-1=b, ede-1=ad-1 >

Subgroups: 652 in 108 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C22 [×2], C22 [×11], S3, C6, C6 [×4], C2×C4 [×3], D4 [×6], C23, C23 [×5], C9, Dic3 [×3], D6 [×3], C2×C6 [×2], C2×C6 [×8], C22⋊C4 [×3], C2×D4 [×3], C24, D9, C18 [×2], C2×Dic3 [×3], C3⋊D4 [×6], C22×S3, C22×C6, C22×C6 [×4], C22≀C2, Dic9, C3.A4, D18, C2×C18, C6.D4 [×3], C2×C3⋊D4 [×3], C23×C6, C9⋊D4, C3.S4, C2×C3.A4, C2×C3.A4, C244S3, C6.S4, C2×C3.S4, C22×C3.A4, C23.D18
Quotients: C1, C2 [×3], C22, S3, D4, D6, D9, C3⋊D4, S4, D18, C2×S4, C9⋊D4, C3.S4, A4⋊D4, C2×C3.S4, C23.D18

Character table of C23.D18

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 18A 18B 18C 18D 18E 18F 18G 18H 18I size 1 1 2 3 3 6 36 2 36 36 36 2 2 2 6 6 6 6 8 8 8 8 8 8 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 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 -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 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 -1 linear of order 2 ρ5 2 2 2 2 2 2 0 2 0 0 0 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 -2 2 2 -2 0 2 0 0 0 -2 2 -2 2 2 -2 -2 -1 -1 -1 -1 1 1 1 1 1 -1 -1 1 orthogonal lifted from D6 ρ7 2 -2 0 -2 2 0 0 2 0 0 0 0 -2 0 -2 2 0 0 2 2 2 -2 0 0 0 0 0 -2 -2 0 orthogonal lifted from D4 ρ8 2 2 2 2 2 2 0 -1 0 0 0 -1 -1 -1 -1 -1 -1 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 orthogonal lifted from D9 ρ9 2 2 2 2 2 2 0 -1 0 0 0 -1 -1 -1 -1 -1 -1 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 orthogonal lifted from D9 ρ10 2 2 -2 2 2 -2 0 -1 0 0 0 1 -1 1 -1 -1 1 1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -ζ95-ζ94 -ζ97-ζ92 -ζ98-ζ9 -ζ95-ζ94 -ζ98-ζ9 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 orthogonal lifted from D18 ρ11 2 2 2 2 2 2 0 -1 0 0 0 -1 -1 -1 -1 -1 -1 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 orthogonal lifted from D9 ρ12 2 2 -2 2 2 -2 0 -1 0 0 0 1 -1 1 -1 -1 1 1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -ζ97-ζ92 -ζ98-ζ9 -ζ95-ζ94 -ζ97-ζ92 -ζ95-ζ94 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 orthogonal lifted from D18 ρ13 2 2 -2 2 2 -2 0 -1 0 0 0 1 -1 1 -1 -1 1 1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -ζ98-ζ9 -ζ95-ζ94 -ζ97-ζ92 -ζ98-ζ9 -ζ97-ζ92 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 orthogonal lifted from D18 ρ14 2 -2 0 -2 2 0 0 -1 0 0 0 -√-3 1 √-3 1 -1 √-3 -√-3 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -ζ97-ζ92 -ζ97+ζ92 -ζ98+ζ9 -ζ95+ζ94 ζ97-ζ92 ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 ζ98-ζ9 complex lifted from C9⋊D4 ρ15 2 -2 0 -2 2 0 0 2 0 0 0 0 -2 0 -2 2 0 0 -1 -1 -1 1 √-3 -√-3 -√-3 -√-3 √-3 1 1 √-3 complex lifted from C3⋊D4 ρ16 2 -2 0 -2 2 0 0 -1 0 0 0 √-3 1 -√-3 1 -1 -√-3 √-3 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -ζ98-ζ9 -ζ98+ζ9 ζ95-ζ94 -ζ97+ζ92 ζ98-ζ9 ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ95+ζ94 complex lifted from C9⋊D4 ρ17 2 -2 0 -2 2 0 0 2 0 0 0 0 -2 0 -2 2 0 0 -1 -1 -1 1 -√-3 √-3 √-3 √-3 -√-3 1 1 -√-3 complex lifted from C3⋊D4 ρ18 2 -2 0 -2 2 0 0 -1 0 0 0 √-3 1 -√-3 1 -1 -√-3 √-3 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -ζ95-ζ94 -ζ95+ζ94 -ζ97+ζ92 ζ98-ζ9 ζ95-ζ94 -ζ98+ζ9 -ζ98-ζ9 -ζ97-ζ92 ζ97-ζ92 complex lifted from C9⋊D4 ρ19 2 -2 0 -2 2 0 0 -1 0 0 0 -√-3 1 √-3 1 -1 √-3 -√-3 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -ζ95-ζ94 ζ95-ζ94 ζ97-ζ92 -ζ98+ζ9 -ζ95+ζ94 ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ97+ζ92 complex lifted from C9⋊D4 ρ20 2 -2 0 -2 2 0 0 -1 0 0 0 √-3 1 -√-3 1 -1 -√-3 √-3 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -ζ97-ζ92 ζ97-ζ92 ζ98-ζ9 ζ95-ζ94 -ζ97+ζ92 -ζ95+ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ98+ζ9 complex lifted from C9⋊D4 ρ21 2 -2 0 -2 2 0 0 -1 0 0 0 -√-3 1 √-3 1 -1 √-3 -√-3 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -ζ98-ζ9 ζ98-ζ9 -ζ95+ζ94 ζ97-ζ92 -ζ98+ζ9 -ζ97+ζ92 -ζ97-ζ92 -ζ95-ζ94 ζ95-ζ94 complex lifted from C9⋊D4 ρ22 3 3 -3 -1 -1 1 -1 3 -1 1 1 -3 3 -3 -1 -1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ23 3 3 3 -1 -1 -1 -1 3 1 1 -1 3 3 3 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ24 3 3 3 -1 -1 -1 1 3 -1 -1 1 3 3 3 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ25 3 3 -3 -1 -1 1 1 3 1 -1 -1 -3 3 -3 -1 -1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ26 6 6 -6 -2 -2 2 0 -3 0 0 0 3 -3 3 1 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C3.S4 ρ27 6 -6 0 2 -2 0 0 6 0 0 0 0 -6 0 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4⋊D4 ρ28 6 6 6 -2 -2 -2 0 -3 0 0 0 -3 -3 -3 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C3.S4 ρ29 6 -6 0 2 -2 0 0 -3 0 0 0 -3√-3 3 3√-3 -1 1 -√-3 √-3 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ30 6 -6 0 2 -2 0 0 -3 0 0 0 3√-3 3 -3√-3 -1 1 √-3 -√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C23.D18
On 36 points
Generators in S36
```(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)
(1 31)(3 33)(4 34)(6 36)(7 19)(9 21)(10 22)(12 24)(13 25)(15 27)(16 28)(18 30)
(1 31)(2 32)(4 34)(5 35)(7 19)(8 20)(10 22)(11 23)(13 25)(14 26)(16 28)(17 29)
(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)
(1 18 31 30)(2 29 32 17)(3 16 33 28)(4 27 34 15)(5 14 35 26)(6 25 36 13)(7 12 19 24)(8 23 20 11)(9 10 21 22)```

`G:=sub<Sym(36)| (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30), (1,31)(3,33)(4,34)(6,36)(7,19)(9,21)(10,22)(12,24)(13,25)(15,27)(16,28)(18,30), (1,31)(2,32)(4,34)(5,35)(7,19)(8,20)(10,22)(11,23)(13,25)(14,26)(16,28)(17,29), (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), (1,18,31,30)(2,29,32,17)(3,16,33,28)(4,27,34,15)(5,14,35,26)(6,25,36,13)(7,12,19,24)(8,23,20,11)(9,10,21,22)>;`

`G:=Group( (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30), (1,31)(3,33)(4,34)(6,36)(7,19)(9,21)(10,22)(12,24)(13,25)(15,27)(16,28)(18,30), (1,31)(2,32)(4,34)(5,35)(7,19)(8,20)(10,22)(11,23)(13,25)(14,26)(16,28)(17,29), (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), (1,18,31,30)(2,29,32,17)(3,16,33,28)(4,27,34,15)(5,14,35,26)(6,25,36,13)(7,12,19,24)(8,23,20,11)(9,10,21,22) );`

`G=PermutationGroup([(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30)], [(1,31),(3,33),(4,34),(6,36),(7,19),(9,21),(10,22),(12,24),(13,25),(15,27),(16,28),(18,30)], [(1,31),(2,32),(4,34),(5,35),(7,19),(8,20),(10,22),(11,23),(13,25),(14,26),(16,28),(17,29)], [(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)], [(1,18,31,30),(2,29,32,17),(3,16,33,28),(4,27,34,15),(5,14,35,26),(6,25,36,13),(7,12,19,24),(8,23,20,11),(9,10,21,22)])`

Matrix representation of C23.D18 in GL5(𝔽37)

 36 0 0 0 0 0 36 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 36 0 1 0 0 36 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 36 1 0 0 0 36 0 0 0 1 36 0
,
 9 21 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 28 16 0 0 0 18 9 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

`G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,36,36,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,36,36,36,0,0,1,0,0],[9,0,0,0,0,21,4,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[28,18,0,0,0,16,9,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;`

C23.D18 in GAP, Magma, Sage, TeX

`C_2^3.D_{18}`
`% in TeX`

`G:=Group("C2^3.D18");`
`// GroupNames label`

`G:=SmallGroup(288,342);`
`// by ID`

`G=gap.SmallGroup(288,342);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,1123,192,1684,6053,782,3534,1350]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^18=1,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*c*e^-1=b*c=c*b,b*e=e*b,d*c*d^-1=b,e*d*e^-1=a*d^-1>;`
`// generators/relations`

Export

׿
×
𝔽