direct product, non-abelian, soluble
Aliases: C5×CSU2(𝔽3), C10.4S4, SL2(𝔽3).C10, Q8.(C5×S3), C2.2(C5×S4), (C5×Q8).2S3, (C5×SL2(𝔽3)).2C2, SmallGroup(240,102)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C5×SL2(𝔽3) — C5×CSU2(𝔽3) |
| SL2(𝔽3) — C5×CSU2(𝔽3) |
Generators and relations for C5×CSU2(𝔽3)
G = < a,b,c,d,e | a5=b4=d3=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=bc, ebe-1=b2c, dcd-1=b, ede-1=d-1 >
Smallest permutation representation of C5×CSU2(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 54 12 42)(2 55 13 43)(3 51 14 44)(4 52 15 45)(5 53 11 41)(6 76 64 23)(7 77 65 24)(8 78 61 25)(9 79 62 21)(10 80 63 22)(16 33 57 74)(17 34 58 75)(18 35 59 71)(19 31 60 72)(20 32 56 73)(26 69 50 40)(27 70 46 36)(28 66 47 37)(29 67 48 38)(30 68 49 39)
(1 66 12 37)(2 67 13 38)(3 68 14 39)(4 69 15 40)(5 70 11 36)(6 60 64 19)(7 56 65 20)(8 57 61 16)(9 58 62 17)(10 59 63 18)(21 75 79 34)(22 71 80 35)(23 72 76 31)(24 73 77 32)(25 74 78 33)(26 45 50 52)(27 41 46 53)(28 42 47 54)(29 43 48 55)(30 44 49 51)
(16 74 25)(17 75 21)(18 71 22)(19 72 23)(20 73 24)(26 52 69)(27 53 70)(28 54 66)(29 55 67)(30 51 68)(31 76 60)(32 77 56)(33 78 57)(34 79 58)(35 80 59)(36 46 41)(37 47 42)(38 48 43)(39 49 44)(40 50 45)
(1 65 12 7)(2 61 13 8)(3 62 14 9)(4 63 15 10)(5 64 11 6)(16 55 57 43)(17 51 58 44)(18 52 59 45)(19 53 60 41)(20 54 56 42)(21 68 79 39)(22 69 80 40)(23 70 76 36)(24 66 77 37)(25 67 78 38)(26 35 50 71)(27 31 46 72)(28 32 47 73)(29 33 48 74)(30 34 49 75)
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), (1,54,12,42)(2,55,13,43)(3,51,14,44)(4,52,15,45)(5,53,11,41)(6,76,64,23)(7,77,65,24)(8,78,61,25)(9,79,62,21)(10,80,63,22)(16,33,57,74)(17,34,58,75)(18,35,59,71)(19,31,60,72)(20,32,56,73)(26,69,50,40)(27,70,46,36)(28,66,47,37)(29,67,48,38)(30,68,49,39), (1,66,12,37)(2,67,13,38)(3,68,14,39)(4,69,15,40)(5,70,11,36)(6,60,64,19)(7,56,65,20)(8,57,61,16)(9,58,62,17)(10,59,63,18)(21,75,79,34)(22,71,80,35)(23,72,76,31)(24,73,77,32)(25,74,78,33)(26,45,50,52)(27,41,46,53)(28,42,47,54)(29,43,48,55)(30,44,49,51), (16,74,25)(17,75,21)(18,71,22)(19,72,23)(20,73,24)(26,52,69)(27,53,70)(28,54,66)(29,55,67)(30,51,68)(31,76,60)(32,77,56)(33,78,57)(34,79,58)(35,80,59)(36,46,41)(37,47,42)(38,48,43)(39,49,44)(40,50,45), (1,65,12,7)(2,61,13,8)(3,62,14,9)(4,63,15,10)(5,64,11,6)(16,55,57,43)(17,51,58,44)(18,52,59,45)(19,53,60,41)(20,54,56,42)(21,68,79,39)(22,69,80,40)(23,70,76,36)(24,66,77,37)(25,67,78,38)(26,35,50,71)(27,31,46,72)(28,32,47,73)(29,33,48,74)(30,34,49,75)>;
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), (1,54,12,42)(2,55,13,43)(3,51,14,44)(4,52,15,45)(5,53,11,41)(6,76,64,23)(7,77,65,24)(8,78,61,25)(9,79,62,21)(10,80,63,22)(16,33,57,74)(17,34,58,75)(18,35,59,71)(19,31,60,72)(20,32,56,73)(26,69,50,40)(27,70,46,36)(28,66,47,37)(29,67,48,38)(30,68,49,39), (1,66,12,37)(2,67,13,38)(3,68,14,39)(4,69,15,40)(5,70,11,36)(6,60,64,19)(7,56,65,20)(8,57,61,16)(9,58,62,17)(10,59,63,18)(21,75,79,34)(22,71,80,35)(23,72,76,31)(24,73,77,32)(25,74,78,33)(26,45,50,52)(27,41,46,53)(28,42,47,54)(29,43,48,55)(30,44,49,51), (16,74,25)(17,75,21)(18,71,22)(19,72,23)(20,73,24)(26,52,69)(27,53,70)(28,54,66)(29,55,67)(30,51,68)(31,76,60)(32,77,56)(33,78,57)(34,79,58)(35,80,59)(36,46,41)(37,47,42)(38,48,43)(39,49,44)(40,50,45), (1,65,12,7)(2,61,13,8)(3,62,14,9)(4,63,15,10)(5,64,11,6)(16,55,57,43)(17,51,58,44)(18,52,59,45)(19,53,60,41)(20,54,56,42)(21,68,79,39)(22,69,80,40)(23,70,76,36)(24,66,77,37)(25,67,78,38)(26,35,50,71)(27,31,46,72)(28,32,47,73)(29,33,48,74)(30,34,49,75) );
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)], [(1,54,12,42),(2,55,13,43),(3,51,14,44),(4,52,15,45),(5,53,11,41),(6,76,64,23),(7,77,65,24),(8,78,61,25),(9,79,62,21),(10,80,63,22),(16,33,57,74),(17,34,58,75),(18,35,59,71),(19,31,60,72),(20,32,56,73),(26,69,50,40),(27,70,46,36),(28,66,47,37),(29,67,48,38),(30,68,49,39)], [(1,66,12,37),(2,67,13,38),(3,68,14,39),(4,69,15,40),(5,70,11,36),(6,60,64,19),(7,56,65,20),(8,57,61,16),(9,58,62,17),(10,59,63,18),(21,75,79,34),(22,71,80,35),(23,72,76,31),(24,73,77,32),(25,74,78,33),(26,45,50,52),(27,41,46,53),(28,42,47,54),(29,43,48,55),(30,44,49,51)], [(16,74,25),(17,75,21),(18,71,22),(19,72,23),(20,73,24),(26,52,69),(27,53,70),(28,54,66),(29,55,67),(30,51,68),(31,76,60),(32,77,56),(33,78,57),(34,79,58),(35,80,59),(36,46,41),(37,47,42),(38,48,43),(39,49,44),(40,50,45)], [(1,65,12,7),(2,61,13,8),(3,62,14,9),(4,63,15,10),(5,64,11,6),(16,55,57,43),(17,51,58,44),(18,52,59,45),(19,53,60,41),(20,54,56,42),(21,68,79,39),(22,69,80,40),(23,70,76,36),(24,66,77,37),(25,67,78,38),(26,35,50,71),(27,31,46,72),(28,32,47,73),(29,33,48,74),(30,34,49,75)]])
C5×CSU2(𝔽3) is a maximal subgroup of
CSU2(𝔽3)⋊D5 Dic5.7S4 D10.2S4
40 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | 30B | 30C | 30D | 40A | ··· | 40H |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 40 | ··· | 40 |
| size | 1 | 1 | 8 | 6 | 12 | 1 | 1 | 1 | 1 | 8 | 6 | 6 | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 8 | 6 | ··· | 6 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 |
| type | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C5 | C10 | S3 | C5×S3 | CSU2(𝔽3) | C5×CSU2(𝔽3) | S4 | C5×S4 | CSU2(𝔽3) | C5×CSU2(𝔽3) |
| kernel | C5×CSU2(𝔽3) | C5×SL2(𝔽3) | CSU2(𝔽3) | SL2(𝔽3) | C5×Q8 | Q8 | C5 | C1 | C10 | C2 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 1 | 4 | 2 | 8 | 2 | 8 | 1 | 4 |
Matrix representation of C5×CSU2(𝔽3) ►in GL2(𝔽31) generated by
| 16 | 0 |
| 0 | 16 |
| 19 | 9 |
| 8 | 12 |
| 28 | 15 |
| 20 | 3 |
| 8 | 11 |
| 30 | 22 |
| 26 | 7 |
| 14 | 5 |
G:=sub<GL(2,GF(31))| [16,0,0,16],[19,8,9,12],[28,20,15,3],[8,30,11,22],[26,14,7,5] >;
C5×CSU2(𝔽3) in GAP, Magma, Sage, TeX
C_5\times {\rm CSU}_2({\mathbb F}_3) % in TeX
G:=Group("C5xCSU(2,3)"); // GroupNames label
G:=SmallGroup(240,102);
// by ID
G=gap.SmallGroup(240,102);
# by ID
G:=PCGroup([6,-2,-5,-3,-2,2,-2,720,362,1443,447,117,904,286,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=d^3=1,c^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*c*e^-1=b^-1,d*b*d^-1=b*c,e*b*e^-1=b^2*c,d*c*d^-1=b,e*d*e^-1=d^-1>;
// generators/relations
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