direct product, non-abelian, soluble
Aliases: C9×SL2(𝔽3), C18.4A4, C2.(C9×A4), Q8⋊C9⋊3C3, Q8⋊1(C3×C9), C6.1(C3×A4), (Q8×C9)⋊1C3, (C3×Q8).1C32, C3.1(C3×SL2(𝔽3)), (C3×SL2(𝔽3)).2C3, SmallGroup(216,38)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C9×SL2(𝔽3) |
| Q8 — C9×SL2(𝔽3) |
Generators and relations for C9×SL2(𝔽3)
G = < a,b,c,d | a9=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >
Smallest permutation representation of C9×SL2(𝔽3)
►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 21 14 64)(2 22 15 65)(3 23 16 66)(4 24 17 67)(5 25 18 68)(6 26 10 69)(7 27 11 70)(8 19 12 71)(9 20 13 72)(28 46 41 60)(29 47 42 61)(30 48 43 62)(31 49 44 63)(32 50 45 55)(33 51 37 56)(34 52 38 57)(35 53 39 58)(36 54 40 59)
(1 58 14 53)(2 59 15 54)(3 60 16 46)(4 61 17 47)(5 62 18 48)(6 63 10 49)(7 55 11 50)(8 56 12 51)(9 57 13 52)(19 37 71 33)(20 38 72 34)(21 39 64 35)(22 40 65 36)(23 41 66 28)(24 42 67 29)(25 43 68 30)(26 44 69 31)(27 45 70 32)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 43 59)(20 44 60)(21 45 61)(22 37 62)(23 38 63)(24 39 55)(25 40 56)(26 41 57)(27 42 58)(28 52 69)(29 53 70)(30 54 71)(31 46 72)(32 47 64)(33 48 65)(34 49 66)(35 50 67)(36 51 68)
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), (1,21,14,64)(2,22,15,65)(3,23,16,66)(4,24,17,67)(5,25,18,68)(6,26,10,69)(7,27,11,70)(8,19,12,71)(9,20,13,72)(28,46,41,60)(29,47,42,61)(30,48,43,62)(31,49,44,63)(32,50,45,55)(33,51,37,56)(34,52,38,57)(35,53,39,58)(36,54,40,59), (1,58,14,53)(2,59,15,54)(3,60,16,46)(4,61,17,47)(5,62,18,48)(6,63,10,49)(7,55,11,50)(8,56,12,51)(9,57,13,52)(19,37,71,33)(20,38,72,34)(21,39,64,35)(22,40,65,36)(23,41,66,28)(24,42,67,29)(25,43,68,30)(26,44,69,31)(27,45,70,32), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,43,59)(20,44,60)(21,45,61)(22,37,62)(23,38,63)(24,39,55)(25,40,56)(26,41,57)(27,42,58)(28,52,69)(29,53,70)(30,54,71)(31,46,72)(32,47,64)(33,48,65)(34,49,66)(35,50,67)(36,51,68)>;
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), (1,21,14,64)(2,22,15,65)(3,23,16,66)(4,24,17,67)(5,25,18,68)(6,26,10,69)(7,27,11,70)(8,19,12,71)(9,20,13,72)(28,46,41,60)(29,47,42,61)(30,48,43,62)(31,49,44,63)(32,50,45,55)(33,51,37,56)(34,52,38,57)(35,53,39,58)(36,54,40,59), (1,58,14,53)(2,59,15,54)(3,60,16,46)(4,61,17,47)(5,62,18,48)(6,63,10,49)(7,55,11,50)(8,56,12,51)(9,57,13,52)(19,37,71,33)(20,38,72,34)(21,39,64,35)(22,40,65,36)(23,41,66,28)(24,42,67,29)(25,43,68,30)(26,44,69,31)(27,45,70,32), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,43,59)(20,44,60)(21,45,61)(22,37,62)(23,38,63)(24,39,55)(25,40,56)(26,41,57)(27,42,58)(28,52,69)(29,53,70)(30,54,71)(31,46,72)(32,47,64)(33,48,65)(34,49,66)(35,50,67)(36,51,68) );
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)], [(1,21,14,64),(2,22,15,65),(3,23,16,66),(4,24,17,67),(5,25,18,68),(6,26,10,69),(7,27,11,70),(8,19,12,71),(9,20,13,72),(28,46,41,60),(29,47,42,61),(30,48,43,62),(31,49,44,63),(32,50,45,55),(33,51,37,56),(34,52,38,57),(35,53,39,58),(36,54,40,59)], [(1,58,14,53),(2,59,15,54),(3,60,16,46),(4,61,17,47),(5,62,18,48),(6,63,10,49),(7,55,11,50),(8,56,12,51),(9,57,13,52),(19,37,71,33),(20,38,72,34),(21,39,64,35),(22,40,65,36),(23,41,66,28),(24,42,67,29),(25,43,68,30),(26,44,69,31),(27,45,70,32)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,43,59),(20,44,60),(21,45,61),(22,37,62),(23,38,63),(24,39,55),(25,40,56),(26,41,57),(27,42,58),(28,52,69),(29,53,70),(30,54,71),(31,46,72),(32,47,64),(33,48,65),(34,49,66),(35,50,67),(36,51,68)]])
C9×SL2(𝔽3) is a maximal subgroup of
C18.5S4 C18.6S4 Dic9.2A4
63 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 4 | 6A | 6B | 6C | ··· | 6H | 9A | ··· | 9F | 9G | ··· | 9R | 12A | 12B | 18A | ··· | 18F | 18G | ··· | 18R | 36A | ··· | 36F |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 6 | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 6 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 |
| type | + | - | + | |||||||||
| image | C1 | C3 | C3 | C3 | C9 | SL2(𝔽3) | SL2(𝔽3) | C3×SL2(𝔽3) | C9×SL2(𝔽3) | A4 | C3×A4 | C9×A4 |
| kernel | C9×SL2(𝔽3) | Q8⋊C9 | Q8×C9 | C3×SL2(𝔽3) | SL2(𝔽3) | C9 | C9 | C3 | C1 | C18 | C6 | C2 |
| # reps | 1 | 4 | 2 | 2 | 18 | 1 | 2 | 6 | 18 | 1 | 2 | 6 |
Matrix representation of C9×SL2(𝔽3) ►in GL2(𝔽19) generated by
| 16 | 0 |
| 0 | 16 |
| 0 | 10 |
| 17 | 0 |
| 12 | 4 |
| 16 | 7 |
| 7 | 9 |
| 0 | 11 |
G:=sub<GL(2,GF(19))| [16,0,0,16],[0,17,10,0],[12,16,4,7],[7,0,9,11] >;
C9×SL2(𝔽3) in GAP, Magma, Sage, TeX
C_9\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C9xSL(2,3)"); // GroupNames label
G:=SmallGroup(216,38);
// by ID
G=gap.SmallGroup(216,38);
# by ID
G:=PCGroup([6,-3,-3,-3,-2,2,-2,43,1299,117,2434,202,88]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^4=d^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=c,d*c*d^-1=b*c>;
// generators/relations
Export