direct product, non-abelian, soluble
Aliases: C32×SL2(𝔽3), Q8⋊C33, C6.8(C3×A4), (C3×C6).6A4, C2.(C32×A4), (C3×Q8)⋊C32, (Q8×C32)⋊3C3, SmallGroup(216,134)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C32×SL2(𝔽3) |
| Q8 — C32×SL2(𝔽3) |
Generators and relations for C32×SL2(𝔽3)
G = < a,b,c,d,e | a3=b3=c4=e3=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >
Subgroups: 234 in 90 conjugacy classes, 40 normal (7 characteristic)
C1, C2, C3, C3, C4, C6, C6, Q8, C32, C32, C12, C3×C6, C3×C6, SL2(𝔽3), C3×Q8, C33, C3×C12, C32×C6, C3×SL2(𝔽3), Q8×C32, C32×SL2(𝔽3)
Quotients: C1, C3, C32, A4, SL2(𝔽3), C33, C3×A4, C3×SL2(𝔽3), C32×A4, C32×SL2(𝔽3)
►On 72 points
Generators in S72
(1 49 25)(2 50 26)(3 51 27)(4 52 28)(5 53 29)(6 54 30)(7 55 31)(8 56 32)(9 57 33)(10 58 34)(11 59 35)(12 60 36)(13 61 37)(14 62 38)(15 63 39)(16 64 40)(17 65 41)(18 66 42)(19 67 43)(20 68 44)(21 69 45)(22 70 46)(23 71 47)(24 72 48)
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(49 65 57)(50 66 58)(51 67 59)(52 68 60)(53 69 61)(54 70 62)(55 71 63)(56 72 64)
(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 7 3 5)(2 6 4 8)(9 15 11 13)(10 14 12 16)(17 23 19 21)(18 22 20 24)(25 31 27 29)(26 30 28 32)(33 39 35 37)(34 38 36 40)(41 47 43 45)(42 46 44 48)(49 55 51 53)(50 54 52 56)(57 63 59 61)(58 62 60 64)(65 71 67 69)(66 70 68 72)
(1 65 33)(2 70 39)(3 67 35)(4 72 37)(5 68 40)(6 71 34)(7 66 38)(8 69 36)(9 49 41)(10 54 47)(11 51 43)(12 56 45)(13 52 48)(14 55 42)(15 50 46)(16 53 44)(17 57 25)(18 62 31)(19 59 27)(20 64 29)(21 60 32)(22 63 26)(23 58 30)(24 61 28)
G:=sub<Sym(72)| (1,49,25)(2,50,26)(3,51,27)(4,52,28)(5,53,29)(6,54,30)(7,55,31)(8,56,32)(9,57,33)(10,58,34)(11,59,35)(12,60,36)(13,61,37)(14,62,38)(15,63,39)(16,64,40)(17,65,41)(18,66,42)(19,67,43)(20,68,44)(21,69,45)(22,70,46)(23,71,47)(24,72,48), (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,65,57)(50,66,58)(51,67,59)(52,68,60)(53,69,61)(54,70,62)(55,71,63)(56,72,64), (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), (1,7,3,5)(2,6,4,8)(9,15,11,13)(10,14,12,16)(17,23,19,21)(18,22,20,24)(25,31,27,29)(26,30,28,32)(33,39,35,37)(34,38,36,40)(41,47,43,45)(42,46,44,48)(49,55,51,53)(50,54,52,56)(57,63,59,61)(58,62,60,64)(65,71,67,69)(66,70,68,72), (1,65,33)(2,70,39)(3,67,35)(4,72,37)(5,68,40)(6,71,34)(7,66,38)(8,69,36)(9,49,41)(10,54,47)(11,51,43)(12,56,45)(13,52,48)(14,55,42)(15,50,46)(16,53,44)(17,57,25)(18,62,31)(19,59,27)(20,64,29)(21,60,32)(22,63,26)(23,58,30)(24,61,28)>;
G:=Group( (1,49,25)(2,50,26)(3,51,27)(4,52,28)(5,53,29)(6,54,30)(7,55,31)(8,56,32)(9,57,33)(10,58,34)(11,59,35)(12,60,36)(13,61,37)(14,62,38)(15,63,39)(16,64,40)(17,65,41)(18,66,42)(19,67,43)(20,68,44)(21,69,45)(22,70,46)(23,71,47)(24,72,48), (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,65,57)(50,66,58)(51,67,59)(52,68,60)(53,69,61)(54,70,62)(55,71,63)(56,72,64), (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), (1,7,3,5)(2,6,4,8)(9,15,11,13)(10,14,12,16)(17,23,19,21)(18,22,20,24)(25,31,27,29)(26,30,28,32)(33,39,35,37)(34,38,36,40)(41,47,43,45)(42,46,44,48)(49,55,51,53)(50,54,52,56)(57,63,59,61)(58,62,60,64)(65,71,67,69)(66,70,68,72), (1,65,33)(2,70,39)(3,67,35)(4,72,37)(5,68,40)(6,71,34)(7,66,38)(8,69,36)(9,49,41)(10,54,47)(11,51,43)(12,56,45)(13,52,48)(14,55,42)(15,50,46)(16,53,44)(17,57,25)(18,62,31)(19,59,27)(20,64,29)(21,60,32)(22,63,26)(23,58,30)(24,61,28) );
G=PermutationGroup([[(1,49,25),(2,50,26),(3,51,27),(4,52,28),(5,53,29),(6,54,30),(7,55,31),(8,56,32),(9,57,33),(10,58,34),(11,59,35),(12,60,36),(13,61,37),(14,62,38),(15,63,39),(16,64,40),(17,65,41),(18,66,42),(19,67,43),(20,68,44),(21,69,45),(22,70,46),(23,71,47),(24,72,48)], [(1,17,9),(2,18,10),(3,19,11),(4,20,12),(5,21,13),(6,22,14),(7,23,15),(8,24,16),(25,41,33),(26,42,34),(27,43,35),(28,44,36),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(49,65,57),(50,66,58),(51,67,59),(52,68,60),(53,69,61),(54,70,62),(55,71,63),(56,72,64)], [(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)], [(1,7,3,5),(2,6,4,8),(9,15,11,13),(10,14,12,16),(17,23,19,21),(18,22,20,24),(25,31,27,29),(26,30,28,32),(33,39,35,37),(34,38,36,40),(41,47,43,45),(42,46,44,48),(49,55,51,53),(50,54,52,56),(57,63,59,61),(58,62,60,64),(65,71,67,69),(66,70,68,72)], [(1,65,33),(2,70,39),(3,67,35),(4,72,37),(5,68,40),(6,71,34),(7,66,38),(8,69,36),(9,49,41),(10,54,47),(11,51,43),(12,56,45),(13,52,48),(14,55,42),(15,50,46),(16,53,44),(17,57,25),(18,62,31),(19,59,27),(20,64,29),(21,60,32),(22,63,26),(23,58,30),(24,61,28)]])
C32×SL2(𝔽3) is a maximal subgroup of
C32⋊4CSU2(𝔽3) C32⋊5GL2(𝔽3) C3⋊Dic3.2A4
63 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3Z | 4 | 6A | ··· | 6H | 6I | ··· | 6Z | 12A | ··· | 12H |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 |
63 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 |
| type | + | - | + | |||||
| image | C1 | C3 | C3 | SL2(𝔽3) | SL2(𝔽3) | C3×SL2(𝔽3) | A4 | C3×A4 |
| kernel | C32×SL2(𝔽3) | C3×SL2(𝔽3) | Q8×C32 | C32 | C32 | C3 | C3×C6 | C6 |
| # reps | 1 | 24 | 2 | 1 | 2 | 24 | 1 | 8 |
Matrix representation of C32×SL2(𝔽3) ►in GL3(𝔽13) generated by
| 3 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 3 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 1 | 0 | 0 |
| 0 | 12 | 2 |
| 0 | 12 | 1 |
| 1 | 0 | 0 |
| 0 | 6 | 6 |
| 0 | 9 | 7 |
| 3 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 12 | 1 |
G:=sub<GL(3,GF(13))| [3,0,0,0,1,0,0,0,1],[3,0,0,0,9,0,0,0,9],[1,0,0,0,12,12,0,2,1],[1,0,0,0,6,9,0,6,7],[3,0,0,0,9,12,0,0,1] >;
C32×SL2(𝔽3) in GAP, Magma, Sage, TeX
C_3^2\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C3^2xSL(2,3)"); // GroupNames label
G:=SmallGroup(216,134);
// by ID
G=gap.SmallGroup(216,134);
# by ID
G:=PCGroup([6,-3,-3,-3,-2,2,-2,1299,117,2434,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^4=e^3=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations