direct product, non-abelian, soluble
Aliases: C5×C4.3S4, C20.9S4, GL2(𝔽3)⋊2C10, C4.3(C5×S4), C4.A4⋊1C10, C10.35(C2×S4), C2.10(C10×S4), (C5×Q8).17D6, Q8.5(S3×C10), (C5×GL2(𝔽3))⋊6C2, SL2(𝔽3)⋊2(C2×C10), (C5×SL2(𝔽3))⋊10C22, C4○D4⋊2(C5×S3), (C5×C4.A4)⋊5C2, (C5×C4○D4)⋊4S3, SmallGroup(480,1021)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C5×SL2(𝔽3) — C5×GL2(𝔽3) — C5×C4.3S4 |
| SL2(𝔽3) — C5×C4.3S4 |
Generators and relations for C5×C4.3S4
G = < a,b,c,d,e,f | a5=b4=e3=f2=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, dcd-1=b2c, ece-1=b2cd, fcf=cd, ede-1=c, fdf=b2d, fef=e-1 >
Subgroups: 346 in 84 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, C23, C10, C10, C12, D6, C15, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, C2×C10, SL2(𝔽3), D12, C5×S3, C30, C8⋊C22, C40, C2×C20, C5×D4, C5×Q8, C22×C10, GL2(𝔽3), C4.A4, C60, S3×C10, C5×M4(2), C5×D8, C5×SD16, D4×C10, C5×C4○D4, C4.3S4, C5×SL2(𝔽3), C5×D12, C5×C8⋊C22, C5×GL2(𝔽3), C5×C4.A4, C5×C4.3S4
Quotients: C1, C2, C22, C5, S3, C10, D6, C2×C10, S4, C5×S3, C2×S4, S3×C10, C4.3S4, C5×S4, C10×S4, C5×C4.3S4
►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 49 12 20)(2 50 13 16)(3 46 14 17)(4 47 15 18)(5 48 11 19)(6 69 80 40)(7 70 76 36)(8 66 77 37)(9 67 78 38)(10 68 79 39)(21 44 62 28)(22 45 63 29)(23 41 64 30)(24 42 65 26)(25 43 61 27)(31 59 72 54)(32 60 73 55)(33 56 74 51)(34 57 75 52)(35 58 71 53)
(1 37 12 66)(2 38 13 67)(3 39 14 68)(4 40 15 69)(5 36 11 70)(6 18 80 47)(7 19 76 48)(8 20 77 49)(9 16 78 50)(10 17 79 46)(21 35 62 71)(22 31 63 72)(23 32 64 73)(24 33 65 74)(25 34 61 75)(26 51 42 56)(27 52 43 57)(28 53 44 58)(29 54 45 59)(30 55 41 60)
(1 60 12 55)(2 56 13 51)(3 57 14 52)(4 58 15 53)(5 59 11 54)(6 62 80 21)(7 63 76 22)(8 64 77 23)(9 65 78 24)(10 61 79 25)(16 33 50 74)(17 34 46 75)(18 35 47 71)(19 31 48 72)(20 32 49 73)(26 38 42 67)(27 39 43 68)(28 40 44 69)(29 36 45 70)(30 37 41 66)
(6 71 21)(7 72 22)(8 73 23)(9 74 24)(10 75 25)(26 38 56)(27 39 57)(28 40 58)(29 36 59)(30 37 60)(31 63 76)(32 64 77)(33 65 78)(34 61 79)(35 62 80)(41 66 55)(42 67 51)(43 68 52)(44 69 53)(45 70 54)
(6 21)(7 22)(8 23)(9 24)(10 25)(16 50)(17 46)(18 47)(19 48)(20 49)(26 67)(27 68)(28 69)(29 70)(30 66)(36 45)(37 41)(38 42)(39 43)(40 44)(51 56)(52 57)(53 58)(54 59)(55 60)(61 79)(62 80)(63 76)(64 77)(65 78)
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), (1,49,12,20)(2,50,13,16)(3,46,14,17)(4,47,15,18)(5,48,11,19)(6,69,80,40)(7,70,76,36)(8,66,77,37)(9,67,78,38)(10,68,79,39)(21,44,62,28)(22,45,63,29)(23,41,64,30)(24,42,65,26)(25,43,61,27)(31,59,72,54)(32,60,73,55)(33,56,74,51)(34,57,75,52)(35,58,71,53), (1,37,12,66)(2,38,13,67)(3,39,14,68)(4,40,15,69)(5,36,11,70)(6,18,80,47)(7,19,76,48)(8,20,77,49)(9,16,78,50)(10,17,79,46)(21,35,62,71)(22,31,63,72)(23,32,64,73)(24,33,65,74)(25,34,61,75)(26,51,42,56)(27,52,43,57)(28,53,44,58)(29,54,45,59)(30,55,41,60), (1,60,12,55)(2,56,13,51)(3,57,14,52)(4,58,15,53)(5,59,11,54)(6,62,80,21)(7,63,76,22)(8,64,77,23)(9,65,78,24)(10,61,79,25)(16,33,50,74)(17,34,46,75)(18,35,47,71)(19,31,48,72)(20,32,49,73)(26,38,42,67)(27,39,43,68)(28,40,44,69)(29,36,45,70)(30,37,41,66), (6,71,21)(7,72,22)(8,73,23)(9,74,24)(10,75,25)(26,38,56)(27,39,57)(28,40,58)(29,36,59)(30,37,60)(31,63,76)(32,64,77)(33,65,78)(34,61,79)(35,62,80)(41,66,55)(42,67,51)(43,68,52)(44,69,53)(45,70,54), (6,21)(7,22)(8,23)(9,24)(10,25)(16,50)(17,46)(18,47)(19,48)(20,49)(26,67)(27,68)(28,69)(29,70)(30,66)(36,45)(37,41)(38,42)(39,43)(40,44)(51,56)(52,57)(53,58)(54,59)(55,60)(61,79)(62,80)(63,76)(64,77)(65,78)>;
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), (1,49,12,20)(2,50,13,16)(3,46,14,17)(4,47,15,18)(5,48,11,19)(6,69,80,40)(7,70,76,36)(8,66,77,37)(9,67,78,38)(10,68,79,39)(21,44,62,28)(22,45,63,29)(23,41,64,30)(24,42,65,26)(25,43,61,27)(31,59,72,54)(32,60,73,55)(33,56,74,51)(34,57,75,52)(35,58,71,53), (1,37,12,66)(2,38,13,67)(3,39,14,68)(4,40,15,69)(5,36,11,70)(6,18,80,47)(7,19,76,48)(8,20,77,49)(9,16,78,50)(10,17,79,46)(21,35,62,71)(22,31,63,72)(23,32,64,73)(24,33,65,74)(25,34,61,75)(26,51,42,56)(27,52,43,57)(28,53,44,58)(29,54,45,59)(30,55,41,60), (1,60,12,55)(2,56,13,51)(3,57,14,52)(4,58,15,53)(5,59,11,54)(6,62,80,21)(7,63,76,22)(8,64,77,23)(9,65,78,24)(10,61,79,25)(16,33,50,74)(17,34,46,75)(18,35,47,71)(19,31,48,72)(20,32,49,73)(26,38,42,67)(27,39,43,68)(28,40,44,69)(29,36,45,70)(30,37,41,66), (6,71,21)(7,72,22)(8,73,23)(9,74,24)(10,75,25)(26,38,56)(27,39,57)(28,40,58)(29,36,59)(30,37,60)(31,63,76)(32,64,77)(33,65,78)(34,61,79)(35,62,80)(41,66,55)(42,67,51)(43,68,52)(44,69,53)(45,70,54), (6,21)(7,22)(8,23)(9,24)(10,25)(16,50)(17,46)(18,47)(19,48)(20,49)(26,67)(27,68)(28,69)(29,70)(30,66)(36,45)(37,41)(38,42)(39,43)(40,44)(51,56)(52,57)(53,58)(54,59)(55,60)(61,79)(62,80)(63,76)(64,77)(65,78) );
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)], [(1,49,12,20),(2,50,13,16),(3,46,14,17),(4,47,15,18),(5,48,11,19),(6,69,80,40),(7,70,76,36),(8,66,77,37),(9,67,78,38),(10,68,79,39),(21,44,62,28),(22,45,63,29),(23,41,64,30),(24,42,65,26),(25,43,61,27),(31,59,72,54),(32,60,73,55),(33,56,74,51),(34,57,75,52),(35,58,71,53)], [(1,37,12,66),(2,38,13,67),(3,39,14,68),(4,40,15,69),(5,36,11,70),(6,18,80,47),(7,19,76,48),(8,20,77,49),(9,16,78,50),(10,17,79,46),(21,35,62,71),(22,31,63,72),(23,32,64,73),(24,33,65,74),(25,34,61,75),(26,51,42,56),(27,52,43,57),(28,53,44,58),(29,54,45,59),(30,55,41,60)], [(1,60,12,55),(2,56,13,51),(3,57,14,52),(4,58,15,53),(5,59,11,54),(6,62,80,21),(7,63,76,22),(8,64,77,23),(9,65,78,24),(10,61,79,25),(16,33,50,74),(17,34,46,75),(18,35,47,71),(19,31,48,72),(20,32,49,73),(26,38,42,67),(27,39,43,68),(28,40,44,69),(29,36,45,70),(30,37,41,66)], [(6,71,21),(7,72,22),(8,73,23),(9,74,24),(10,75,25),(26,38,56),(27,39,57),(28,40,58),(29,36,59),(30,37,60),(31,63,76),(32,64,77),(33,65,78),(34,61,79),(35,62,80),(41,66,55),(42,67,51),(43,68,52),(44,69,53),(45,70,54)], [(6,21),(7,22),(8,23),(9,24),(10,25),(16,50),(17,46),(18,47),(19,48),(20,49),(26,67),(27,68),(28,69),(29,70),(30,66),(36,45),(37,41),(38,42),(39,43),(40,44),(51,56),(52,57),(53,58),(54,59),(55,60),(61,79),(62,80),(63,76),(64,77),(65,78)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 12 | 12 | 8 | 2 | 6 | 1 | 1 | 1 | 1 | 8 | 12 | 12 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 8 | 8 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 8 | ··· | 8 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D6 | C5×S3 | S3×C10 | S4 | C2×S4 | C5×S4 | C10×S4 | C4.3S4 | C5×C4.3S4 |
| kernel | C5×C4.3S4 | C5×GL2(𝔽3) | C5×C4.A4 | C4.3S4 | GL2(𝔽3) | C4.A4 | C5×C4○D4 | C5×Q8 | C4○D4 | Q8 | C20 | C10 | C4 | C2 | C5 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 | 3 | 12 |
Matrix representation of C5×C4.3S4 ►in GL4(𝔽241) generated by
| 91 | 0 | 0 | 0 |
| 0 | 91 | 0 | 0 |
| 0 | 0 | 91 | 0 |
| 0 | 0 | 0 | 91 |
| 99 | 0 | 198 | 198 |
| 198 | 99 | 0 | 198 |
| 0 | 198 | 99 | 198 |
| 43 | 43 | 43 | 185 |
| 0 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 1 | 1 | 1 | 2 |
| 0 | 240 | 240 | 240 |
| 240 | 240 | 240 | 239 |
| 0 | 0 | 240 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 1 | 1 | 1 | 2 |
| 240 | 240 | 0 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 240 | 240 | 240 | 240 |
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,91,0,0,0,0,91],[99,198,0,43,0,99,198,43,198,0,99,43,198,198,198,185],[0,240,1,0,1,0,1,240,0,0,1,240,0,0,2,240],[240,0,0,1,240,0,1,0,240,240,0,1,239,0,0,1],[1,0,1,240,0,0,1,240,0,240,1,0,0,0,2,240],[1,0,0,240,0,0,1,240,0,1,0,240,0,0,0,240] >;
C5×C4.3S4 in GAP, Magma, Sage, TeX
C_5\times C_4._3S_4
% in TeX
G:=Group("C5xC4.3S4"); // GroupNames label
G:=SmallGroup(480,1021);
// by ID
G=gap.SmallGroup(480,1021);
# by ID
G:=PCGroup([7,-2,-2,-5,-3,-2,2,-2,3389,1688,1123,4204,655,172,2525,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^4=e^3=f^2=1,c^2=d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,d*c*d^-1=b^2*c,e*c*e^-1=b^2*c*d,f*c*f=c*d,e*d*e^-1=c,f*d*f=b^2*d,f*e*f=e^-1>;
// generators/relations