direct product, non-abelian, soluble
Aliases: C10×SL2(𝔽3), Q8⋊C30, (C2×Q8)⋊C15, (Q8×C10)⋊C3, (C5×Q8)⋊2C6, C10.5(C2×A4), C2.2(C10×A4), (C2×C10).2A4, C22.2(C5×A4), SmallGroup(240,153)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — C10×SL2(𝔽3) |
| Q8 — C10×SL2(𝔽3) |
Generators and relations for C10×SL2(𝔽3)
G = < a,b,c,d | a10=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 C10×SL2(𝔽3)
►On 80 points
Generators in S80
(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 73 74 75 76 77 78 79 80)
(1 32 61 77)(2 33 62 78)(3 34 63 79)(4 35 64 80)(5 36 65 71)(6 37 66 72)(7 38 67 73)(8 39 68 74)(9 40 69 75)(10 31 70 76)(11 41 24 54)(12 42 25 55)(13 43 26 56)(14 44 27 57)(15 45 28 58)(16 46 29 59)(17 47 30 60)(18 48 21 51)(19 49 22 52)(20 50 23 53)
(1 18 61 21)(2 19 62 22)(3 20 63 23)(4 11 64 24)(5 12 65 25)(6 13 66 26)(7 14 67 27)(8 15 68 28)(9 16 69 29)(10 17 70 30)(31 60 76 47)(32 51 77 48)(33 52 78 49)(34 53 79 50)(35 54 80 41)(36 55 71 42)(37 56 72 43)(38 57 73 44)(39 58 74 45)(40 59 75 46)
(11 35 54)(12 36 55)(13 37 56)(14 38 57)(15 39 58)(16 40 59)(17 31 60)(18 32 51)(19 33 52)(20 34 53)(21 77 48)(22 78 49)(23 79 50)(24 80 41)(25 71 42)(26 72 43)(27 73 44)(28 74 45)(29 75 46)(30 76 47)
G:=sub<Sym(80)| (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)(71,72,73,74,75,76,77,78,79,80), (1,32,61,77)(2,33,62,78)(3,34,63,79)(4,35,64,80)(5,36,65,71)(6,37,66,72)(7,38,67,73)(8,39,68,74)(9,40,69,75)(10,31,70,76)(11,41,24,54)(12,42,25,55)(13,43,26,56)(14,44,27,57)(15,45,28,58)(16,46,29,59)(17,47,30,60)(18,48,21,51)(19,49,22,52)(20,50,23,53), (1,18,61,21)(2,19,62,22)(3,20,63,23)(4,11,64,24)(5,12,65,25)(6,13,66,26)(7,14,67,27)(8,15,68,28)(9,16,69,29)(10,17,70,30)(31,60,76,47)(32,51,77,48)(33,52,78,49)(34,53,79,50)(35,54,80,41)(36,55,71,42)(37,56,72,43)(38,57,73,44)(39,58,74,45)(40,59,75,46), (11,35,54)(12,36,55)(13,37,56)(14,38,57)(15,39,58)(16,40,59)(17,31,60)(18,32,51)(19,33,52)(20,34,53)(21,77,48)(22,78,49)(23,79,50)(24,80,41)(25,71,42)(26,72,43)(27,73,44)(28,74,45)(29,75,46)(30,76,47)>;
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)(71,72,73,74,75,76,77,78,79,80), (1,32,61,77)(2,33,62,78)(3,34,63,79)(4,35,64,80)(5,36,65,71)(6,37,66,72)(7,38,67,73)(8,39,68,74)(9,40,69,75)(10,31,70,76)(11,41,24,54)(12,42,25,55)(13,43,26,56)(14,44,27,57)(15,45,28,58)(16,46,29,59)(17,47,30,60)(18,48,21,51)(19,49,22,52)(20,50,23,53), (1,18,61,21)(2,19,62,22)(3,20,63,23)(4,11,64,24)(5,12,65,25)(6,13,66,26)(7,14,67,27)(8,15,68,28)(9,16,69,29)(10,17,70,30)(31,60,76,47)(32,51,77,48)(33,52,78,49)(34,53,79,50)(35,54,80,41)(36,55,71,42)(37,56,72,43)(38,57,73,44)(39,58,74,45)(40,59,75,46), (11,35,54)(12,36,55)(13,37,56)(14,38,57)(15,39,58)(16,40,59)(17,31,60)(18,32,51)(19,33,52)(20,34,53)(21,77,48)(22,78,49)(23,79,50)(24,80,41)(25,71,42)(26,72,43)(27,73,44)(28,74,45)(29,75,46)(30,76,47) );
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),(71,72,73,74,75,76,77,78,79,80)], [(1,32,61,77),(2,33,62,78),(3,34,63,79),(4,35,64,80),(5,36,65,71),(6,37,66,72),(7,38,67,73),(8,39,68,74),(9,40,69,75),(10,31,70,76),(11,41,24,54),(12,42,25,55),(13,43,26,56),(14,44,27,57),(15,45,28,58),(16,46,29,59),(17,47,30,60),(18,48,21,51),(19,49,22,52),(20,50,23,53)], [(1,18,61,21),(2,19,62,22),(3,20,63,23),(4,11,64,24),(5,12,65,25),(6,13,66,26),(7,14,67,27),(8,15,68,28),(9,16,69,29),(10,17,70,30),(31,60,76,47),(32,51,77,48),(33,52,78,49),(34,53,79,50),(35,54,80,41),(36,55,71,42),(37,56,72,43),(38,57,73,44),(39,58,74,45),(40,59,75,46)], [(11,35,54),(12,36,55),(13,37,56),(14,38,57),(15,39,58),(16,40,59),(17,31,60),(18,32,51),(19,33,52),(20,34,53),(21,77,48),(22,78,49),(23,79,50),(24,80,41),(25,71,42),(26,72,43),(27,73,44),(28,74,45),(29,75,46),(30,76,47)]])
C10×SL2(𝔽3) is a maximal subgroup of
Q8⋊Dic15 Q8.D30 SL2(𝔽3).11D10
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 10A | ··· | 10L | 15A | ··· | 15H | 20A | ··· | 20H | 30A | ··· | 30X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | + | - | + | + | ||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | SL2(𝔽3) | SL2(𝔽3) | C5×SL2(𝔽3) | A4 | C2×A4 | C5×A4 | C10×A4 |
| kernel | C10×SL2(𝔽3) | C5×SL2(𝔽3) | Q8×C10 | C2×SL2(𝔽3) | C5×Q8 | SL2(𝔽3) | C2×Q8 | Q8 | C10 | C10 | C2 | C2×C10 | C10 | C22 | C2 |
| # reps | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 2 | 4 | 24 | 1 | 1 | 4 | 4 |
Matrix representation of C10×SL2(𝔽3) ►in GL3(𝔽61) generated by
| 60 | 0 | 0 |
| 0 | 52 | 0 |
| 0 | 0 | 52 |
| 1 | 0 | 0 |
| 0 | 0 | 60 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 14 | 13 |
| 0 | 13 | 47 |
| 13 | 0 | 0 |
| 0 | 1 | 13 |
| 0 | 0 | 47 |
G:=sub<GL(3,GF(61))| [60,0,0,0,52,0,0,0,52],[1,0,0,0,0,1,0,60,0],[1,0,0,0,14,13,0,13,47],[13,0,0,0,1,0,0,13,47] >;
C10×SL2(𝔽3) in GAP, Magma, Sage, TeX
C_{10}\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C10xSL(2,3)"); // GroupNames label
G:=SmallGroup(240,153);
// by ID
G=gap.SmallGroup(240,153);
# by ID
G:=PCGroup([6,-2,-3,-5,-2,2,-2,729,117,1360,202,88]);
// Polycyclic
G:=Group<a,b,c,d|a^10=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
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