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## G = D18.A4order 432 = 24·33

### The non-split extension by D18 of A4 acting via A4/C22=C3

Aliases: D18.A4, D9⋊SL2(𝔽3), C6.3(S3×A4), (Q8×C9)⋊4C6, (Q8×D9)⋊2C3, C18.A42C2, C18.2(C2×A4), Q82(C9⋊C6), C9⋊(C2×SL2(𝔽3)), C2.3(D9⋊A4), C3.1(S3×SL2(𝔽3)), (C3×SL2(𝔽3)).2S3, (C3×Q8).13(C3×S3), SmallGroup(432,263)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C9 — D18.A4
 Chief series C1 — C3 — C6 — C18 — Q8×C9 — C18.A4 — D18.A4
 Lower central Q8×C9 — D18.A4
 Upper central C1 — C2

Generators and relations for D18.A4
G = < a,b,c,d,e | a18=b2=e3=1, c2=d2=a9, bab=a-1, ac=ca, ad=da, eae-1=a13, bc=cb, bd=db, ebe-1=a12b, dcd-1=a9c, ece-1=a9cd, ede-1=c >

Subgroups: 419 in 57 conjugacy classes, 16 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, Q8, Q8, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×Q8, D9, C18, C18, C3×S3, C3×C6, SL2(𝔽3), Dic6, C4×S3, C3×Q8, 3- 1+2, Dic9, C36, D18, S3×C6, C2×SL2(𝔽3), S3×Q8, C9⋊C6, C2×3- 1+2, Q8⋊C9, Dic18, C4×D9, Q8×C9, C3×SL2(𝔽3), C2×C9⋊C6, Q8×D9, S3×SL2(𝔽3), C18.A4, D18.A4
Quotients: C1, C2, C3, S3, C6, A4, C3×S3, SL2(𝔽3), C2×A4, C2×SL2(𝔽3), C9⋊C6, S3×A4, S3×SL2(𝔽3), D9⋊A4, D18.A4

Character table of D18.A4

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 12 18A 18B 18C 36A 36B 36C size 1 1 9 9 2 12 12 6 54 2 12 12 36 36 36 36 6 24 24 12 6 24 24 12 12 12 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 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 linear of order 2 ρ3 1 1 -1 -1 1 ζ32 ζ3 1 -1 1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 1 ζ32 ζ3 1 1 ζ32 ζ3 1 1 1 linear of order 6 ρ4 1 1 -1 -1 1 ζ3 ζ32 1 -1 1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 1 ζ3 ζ32 1 1 ζ3 ζ32 1 1 1 linear of order 6 ρ5 1 1 1 1 1 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 ζ32 ζ3 1 1 ζ32 ζ3 1 1 1 linear of order 3 ρ6 1 1 1 1 1 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 ζ3 ζ32 1 1 ζ3 ζ32 1 1 1 linear of order 3 ρ7 2 2 0 0 2 2 2 2 0 2 2 2 0 0 0 0 -1 -1 -1 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 -2 -2 2 2 -1 -1 0 0 -2 1 1 1 1 -1 -1 2 -1 -1 0 -2 1 1 0 0 0 symplectic lifted from SL2(𝔽3), Schur index 2 ρ9 2 -2 2 -2 2 -1 -1 0 0 -2 1 1 -1 -1 1 1 2 -1 -1 0 -2 1 1 0 0 0 symplectic lifted from SL2(𝔽3), Schur index 2 ρ10 2 -2 2 -2 2 ζ65 ζ6 0 0 -2 ζ32 ζ3 ζ65 ζ6 ζ3 ζ32 2 ζ65 ζ6 0 -2 ζ3 ζ32 0 0 0 complex lifted from SL2(𝔽3) ρ11 2 2 0 0 2 -1-√-3 -1+√-3 2 0 2 -1+√-3 -1-√-3 0 0 0 0 -1 ζ6 ζ65 2 -1 ζ6 ζ65 -1 -1 -1 complex lifted from C3×S3 ρ12 2 2 0 0 2 -1+√-3 -1-√-3 2 0 2 -1-√-3 -1+√-3 0 0 0 0 -1 ζ65 ζ6 2 -1 ζ65 ζ6 -1 -1 -1 complex lifted from C3×S3 ρ13 2 -2 -2 2 2 ζ65 ζ6 0 0 -2 ζ32 ζ3 ζ3 ζ32 ζ65 ζ6 2 ζ65 ζ6 0 -2 ζ3 ζ32 0 0 0 complex lifted from SL2(𝔽3) ρ14 2 -2 -2 2 2 ζ6 ζ65 0 0 -2 ζ3 ζ32 ζ32 ζ3 ζ6 ζ65 2 ζ6 ζ65 0 -2 ζ32 ζ3 0 0 0 complex lifted from SL2(𝔽3) ρ15 2 -2 2 -2 2 ζ6 ζ65 0 0 -2 ζ3 ζ32 ζ6 ζ65 ζ32 ζ3 2 ζ6 ζ65 0 -2 ζ32 ζ3 0 0 0 complex lifted from SL2(𝔽3) ρ16 3 3 -3 -3 3 0 0 -1 1 3 0 0 0 0 0 0 3 0 0 -1 3 0 0 -1 -1 -1 orthogonal lifted from C2×A4 ρ17 3 3 3 3 3 0 0 -1 -1 3 0 0 0 0 0 0 3 0 0 -1 3 0 0 -1 -1 -1 orthogonal lifted from A4 ρ18 4 -4 0 0 4 -2 -2 0 0 -4 2 2 0 0 0 0 -2 1 1 0 2 -1 -1 0 0 0 symplectic lifted from S3×SL2(𝔽3), Schur index 2 ρ19 4 -4 0 0 4 1-√-3 1+√-3 0 0 -4 -1-√-3 -1+√-3 0 0 0 0 -2 ζ3 ζ32 0 2 ζ65 ζ6 0 0 0 complex lifted from S3×SL2(𝔽3) ρ20 4 -4 0 0 4 1+√-3 1-√-3 0 0 -4 -1+√-3 -1-√-3 0 0 0 0 -2 ζ32 ζ3 0 2 ζ6 ζ65 0 0 0 complex lifted from S3×SL2(𝔽3) ρ21 6 6 0 0 -3 0 0 6 0 -3 0 0 0 0 0 0 0 0 0 -3 0 0 0 0 0 0 orthogonal lifted from C9⋊C6 ρ22 6 6 0 0 6 0 0 -2 0 6 0 0 0 0 0 0 -3 0 0 -2 -3 0 0 1 1 1 orthogonal lifted from S3×A4 ρ23 6 6 0 0 -3 0 0 -2 0 -3 0 0 0 0 0 0 0 0 0 1 0 0 0 2ζ95+2ζ94 2ζ97+2ζ92 2ζ98+2ζ9 orthogonal lifted from D9⋊A4 ρ24 6 6 0 0 -3 0 0 -2 0 -3 0 0 0 0 0 0 0 0 0 1 0 0 0 2ζ98+2ζ9 2ζ95+2ζ94 2ζ97+2ζ92 orthogonal lifted from D9⋊A4 ρ25 6 6 0 0 -3 0 0 -2 0 -3 0 0 0 0 0 0 0 0 0 1 0 0 0 2ζ97+2ζ92 2ζ98+2ζ9 2ζ95+2ζ94 orthogonal lifted from D9⋊A4 ρ26 12 -12 0 0 -6 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of D18.A4
On 72 points
Generators in S72
```(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 65 66 67 68 69 70 71 72)
(1 9)(2 8)(3 7)(4 6)(10 18)(11 17)(12 16)(13 15)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(25 27)(34 36)(37 45)(38 44)(39 43)(40 42)(46 54)(47 53)(48 52)(49 51)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(68 72)(69 71)
(1 22 10 31)(2 23 11 32)(3 24 12 33)(4 25 13 34)(5 26 14 35)(6 27 15 36)(7 28 16 19)(8 29 17 20)(9 30 18 21)(37 66 46 57)(38 67 47 58)(39 68 48 59)(40 69 49 60)(41 70 50 61)(42 71 51 62)(43 72 52 63)(44 55 53 64)(45 56 54 65)
(1 46 10 37)(2 47 11 38)(3 48 12 39)(4 49 13 40)(5 50 14 41)(6 51 15 42)(7 52 16 43)(8 53 17 44)(9 54 18 45)(19 63 28 72)(20 64 29 55)(21 65 30 56)(22 66 31 57)(23 67 32 58)(24 68 33 59)(25 69 34 60)(26 70 35 61)(27 71 36 62)
(2 8 14)(3 15 9)(5 11 17)(6 18 12)(19 43 72)(20 50 67)(21 39 62)(22 46 57)(23 53 70)(24 42 65)(25 49 60)(26 38 55)(27 45 68)(28 52 63)(29 41 58)(30 48 71)(31 37 66)(32 44 61)(33 51 56)(34 40 69)(35 47 64)(36 54 59)```

`G:=sub<Sym(72)| (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)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,9)(2,8)(3,7)(4,6)(10,18)(11,17)(12,16)(13,15)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)(34,36)(37,45)(38,44)(39,43)(40,42)(46,54)(47,53)(48,52)(49,51)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(68,72)(69,71), (1,22,10,31)(2,23,11,32)(3,24,12,33)(4,25,13,34)(5,26,14,35)(6,27,15,36)(7,28,16,19)(8,29,17,20)(9,30,18,21)(37,66,46,57)(38,67,47,58)(39,68,48,59)(40,69,49,60)(41,70,50,61)(42,71,51,62)(43,72,52,63)(44,55,53,64)(45,56,54,65), (1,46,10,37)(2,47,11,38)(3,48,12,39)(4,49,13,40)(5,50,14,41)(6,51,15,42)(7,52,16,43)(8,53,17,44)(9,54,18,45)(19,63,28,72)(20,64,29,55)(21,65,30,56)(22,66,31,57)(23,67,32,58)(24,68,33,59)(25,69,34,60)(26,70,35,61)(27,71,36,62), (2,8,14)(3,15,9)(5,11,17)(6,18,12)(19,43,72)(20,50,67)(21,39,62)(22,46,57)(23,53,70)(24,42,65)(25,49,60)(26,38,55)(27,45,68)(28,52,63)(29,41,58)(30,48,71)(31,37,66)(32,44,61)(33,51,56)(34,40,69)(35,47,64)(36,54,59)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,9)(2,8)(3,7)(4,6)(10,18)(11,17)(12,16)(13,15)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)(34,36)(37,45)(38,44)(39,43)(40,42)(46,54)(47,53)(48,52)(49,51)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(68,72)(69,71), (1,22,10,31)(2,23,11,32)(3,24,12,33)(4,25,13,34)(5,26,14,35)(6,27,15,36)(7,28,16,19)(8,29,17,20)(9,30,18,21)(37,66,46,57)(38,67,47,58)(39,68,48,59)(40,69,49,60)(41,70,50,61)(42,71,51,62)(43,72,52,63)(44,55,53,64)(45,56,54,65), (1,46,10,37)(2,47,11,38)(3,48,12,39)(4,49,13,40)(5,50,14,41)(6,51,15,42)(7,52,16,43)(8,53,17,44)(9,54,18,45)(19,63,28,72)(20,64,29,55)(21,65,30,56)(22,66,31,57)(23,67,32,58)(24,68,33,59)(25,69,34,60)(26,70,35,61)(27,71,36,62), (2,8,14)(3,15,9)(5,11,17)(6,18,12)(19,43,72)(20,50,67)(21,39,62)(22,46,57)(23,53,70)(24,42,65)(25,49,60)(26,38,55)(27,45,68)(28,52,63)(29,41,58)(30,48,71)(31,37,66)(32,44,61)(33,51,56)(34,40,69)(35,47,64)(36,54,59) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,9),(2,8),(3,7),(4,6),(10,18),(11,17),(12,16),(13,15),(19,33),(20,32),(21,31),(22,30),(23,29),(24,28),(25,27),(34,36),(37,45),(38,44),(39,43),(40,42),(46,54),(47,53),(48,52),(49,51),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62),(68,72),(69,71)], [(1,22,10,31),(2,23,11,32),(3,24,12,33),(4,25,13,34),(5,26,14,35),(6,27,15,36),(7,28,16,19),(8,29,17,20),(9,30,18,21),(37,66,46,57),(38,67,47,58),(39,68,48,59),(40,69,49,60),(41,70,50,61),(42,71,51,62),(43,72,52,63),(44,55,53,64),(45,56,54,65)], [(1,46,10,37),(2,47,11,38),(3,48,12,39),(4,49,13,40),(5,50,14,41),(6,51,15,42),(7,52,16,43),(8,53,17,44),(9,54,18,45),(19,63,28,72),(20,64,29,55),(21,65,30,56),(22,66,31,57),(23,67,32,58),(24,68,33,59),(25,69,34,60),(26,70,35,61),(27,71,36,62)], [(2,8,14),(3,15,9),(5,11,17),(6,18,12),(19,43,72),(20,50,67),(21,39,62),(22,46,57),(23,53,70),(24,42,65),(25,49,60),(26,38,55),(27,45,68),(28,52,63),(29,41,58),(30,48,71),(31,37,66),(32,44,61),(33,51,56),(34,40,69),(35,47,64),(36,54,59)]])`

Matrix representation of D18.A4 in GL10(𝔽37)

 0 36 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 36 36 36 36 35 0 0 0 0 0 0 0 0 1 36 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 1
,
 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 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 0 0 0 0 36 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 0 0 0 36 0 36 36 36 0 0 0 0 1 0 1 0 0 1
,
 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 11 0 10 0 0 0 0 0 0 0 0 11 0 10 0 0 0 0 0 0 10 0 26 0 0 0 0 0 0 0 0 10 0 26 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 10 0 26 0 0 0 0 0 0 0 0 10 0 26 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 1 36 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 36 36 36 0 36 36

`G:=sub<GL(10,GF(37))| [0,1,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,0,0,0,0,0,0,36,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,36,36,0,1,1,0,0,0,0,1,0,36,0,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,0,0,35,36,1,1],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,36,36,0,1,0,0,0,0,0,0,0,1,36,0,0,0,0,0,36,36,0,0,0,1,0,0,0,0,0,1,0,0,36,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,36,1],[0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[11,0,10,0,0,0,0,0,0,0,0,11,0,10,0,0,0,0,0,0,10,0,26,0,0,0,0,0,0,0,0,10,0,26,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,10,0,0,0,0,0,0,0,0,1,0,10,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,36,0,0,0,0,0,1,0,0,0,36,0,0,0,0,0,0,0,1,0,36,0,0,0,0,0,0,36,36,1,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,36] >;`

D18.A4 in GAP, Magma, Sage, TeX

`D_{18}.A_4`
`% in TeX`

`G:=Group("D18.A4");`
`// GroupNames label`

`G:=SmallGroup(432,263);`
`// by ID`

`G=gap.SmallGroup(432,263);`
`# by ID`

`G:=PCGroup([7,-2,-3,-2,2,-3,-2,-3,198,268,94,409,192,6724,2951,452,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^18=b^2=e^3=1,c^2=d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^13,b*c=c*b,b*d=d*b,e*b*e^-1=a^12*b,d*c*d^-1=a^9*c,e*c*e^-1=a^9*c*d,e*d*e^-1=c>;`
`// generators/relations`

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