Aliases: C20.3S4, Q8.5D30, SL2(𝔽3)⋊4D10, C4.3(C5⋊S4), C4.A4⋊1D5, C4○D4⋊2D15, Q8⋊D15⋊2C2, C10.24(C2×S4), C5⋊2(C4.3S4), (C5×Q8).12D6, (C5×SL2(𝔽3))⋊4C22, C2.10(C2×C5⋊S4), (C5×C4○D4)⋊2S3, (C5×C4.A4)⋊1C2, SmallGroup(480,1032)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×SL2(𝔽3) — C20.3S4 |
| C5×SL2(𝔽3) — C20.3S4 |
Subgroups: 890 in 84 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2 [×3], C3, C4, C4, C22 [×5], C5, S3 [×2], C6, C8 [×2], C2×C4, D4 [×4], Q8, C23, D5 [×2], C10, C10, C12, D6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20, C20, D10 [×4], C2×C10, SL2(𝔽3), D12, D15 [×2], C30, C8⋊C22, C5⋊2C8 [×2], D20 [×3], C2×C20, C5×D4, C5×Q8, C22×D5, GL2(𝔽3) [×2], C4.A4, C60, D30 [×2], C4.Dic5, D4⋊D5 [×2], Q8⋊D5 [×2], C2×D20, C5×C4○D4, C4.3S4, C5×SL2(𝔽3), D60, D4⋊D10, Q8⋊D15 [×2], C5×C4.A4, C20.3S4
Quotients:
C1, C2 [×3], C22, S3, D5, D6, D10, S4, D15, C2×S4, D30, C4.3S4, C5⋊S4, C2×C5⋊S4, C20.3S4
Generators and relations
G = < a,b,c,d,e | a20=d3=e2=1, b2=c2=a10, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=a10b, dbd-1=a10bc, ebe=bc, dcd-1=b, ece=a10c, ede=d-1 >
►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 79 11 69)(2 80 12 70)(3 61 13 71)(4 62 14 72)(5 63 15 73)(6 64 16 74)(7 65 17 75)(8 66 18 76)(9 67 19 77)(10 68 20 78)(21 53 31 43)(22 54 32 44)(23 55 33 45)(24 56 34 46)(25 57 35 47)(26 58 36 48)(27 59 37 49)(28 60 38 50)(29 41 39 51)(30 42 40 52)
(1 33 11 23)(2 34 12 24)(3 35 13 25)(4 36 14 26)(5 37 15 27)(6 38 16 28)(7 39 17 29)(8 40 18 30)(9 21 19 31)(10 22 20 32)(41 75 51 65)(42 76 52 66)(43 77 53 67)(44 78 54 68)(45 79 55 69)(46 80 56 70)(47 61 57 71)(48 62 58 72)(49 63 59 73)(50 64 60 74)
(21 53 67)(22 54 68)(23 55 69)(24 56 70)(25 57 71)(26 58 72)(27 59 73)(28 60 74)(29 41 75)(30 42 76)(31 43 77)(32 44 78)(33 45 79)(34 46 80)(35 47 61)(36 48 62)(37 49 63)(38 50 64)(39 51 65)(40 52 66)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(36 40)(37 39)(41 73)(42 72)(43 71)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)(51 63)(52 62)(53 61)(54 80)(55 79)(56 78)(57 77)(58 76)(59 75)(60 74)
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), (1,79,11,69)(2,80,12,70)(3,61,13,71)(4,62,14,72)(5,63,15,73)(6,64,16,74)(7,65,17,75)(8,66,18,76)(9,67,19,77)(10,68,20,78)(21,53,31,43)(22,54,32,44)(23,55,33,45)(24,56,34,46)(25,57,35,47)(26,58,36,48)(27,59,37,49)(28,60,38,50)(29,41,39,51)(30,42,40,52), (1,33,11,23)(2,34,12,24)(3,35,13,25)(4,36,14,26)(5,37,15,27)(6,38,16,28)(7,39,17,29)(8,40,18,30)(9,21,19,31)(10,22,20,32)(41,75,51,65)(42,76,52,66)(43,77,53,67)(44,78,54,68)(45,79,55,69)(46,80,56,70)(47,61,57,71)(48,62,58,72)(49,63,59,73)(50,64,60,74), (21,53,67)(22,54,68)(23,55,69)(24,56,70)(25,57,71)(26,58,72)(27,59,73)(28,60,74)(29,41,75)(30,42,76)(31,43,77)(32,44,78)(33,45,79)(34,46,80)(35,47,61)(36,48,62)(37,49,63)(38,50,64)(39,51,65)(40,52,66), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(36,40)(37,39)(41,73)(42,72)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,80)(55,79)(56,78)(57,77)(58,76)(59,75)(60,74)>;
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), (1,79,11,69)(2,80,12,70)(3,61,13,71)(4,62,14,72)(5,63,15,73)(6,64,16,74)(7,65,17,75)(8,66,18,76)(9,67,19,77)(10,68,20,78)(21,53,31,43)(22,54,32,44)(23,55,33,45)(24,56,34,46)(25,57,35,47)(26,58,36,48)(27,59,37,49)(28,60,38,50)(29,41,39,51)(30,42,40,52), (1,33,11,23)(2,34,12,24)(3,35,13,25)(4,36,14,26)(5,37,15,27)(6,38,16,28)(7,39,17,29)(8,40,18,30)(9,21,19,31)(10,22,20,32)(41,75,51,65)(42,76,52,66)(43,77,53,67)(44,78,54,68)(45,79,55,69)(46,80,56,70)(47,61,57,71)(48,62,58,72)(49,63,59,73)(50,64,60,74), (21,53,67)(22,54,68)(23,55,69)(24,56,70)(25,57,71)(26,58,72)(27,59,73)(28,60,74)(29,41,75)(30,42,76)(31,43,77)(32,44,78)(33,45,79)(34,46,80)(35,47,61)(36,48,62)(37,49,63)(38,50,64)(39,51,65)(40,52,66), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(36,40)(37,39)(41,73)(42,72)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,80)(55,79)(56,78)(57,77)(58,76)(59,75)(60,74) );
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)], [(1,79,11,69),(2,80,12,70),(3,61,13,71),(4,62,14,72),(5,63,15,73),(6,64,16,74),(7,65,17,75),(8,66,18,76),(9,67,19,77),(10,68,20,78),(21,53,31,43),(22,54,32,44),(23,55,33,45),(24,56,34,46),(25,57,35,47),(26,58,36,48),(27,59,37,49),(28,60,38,50),(29,41,39,51),(30,42,40,52)], [(1,33,11,23),(2,34,12,24),(3,35,13,25),(4,36,14,26),(5,37,15,27),(6,38,16,28),(7,39,17,29),(8,40,18,30),(9,21,19,31),(10,22,20,32),(41,75,51,65),(42,76,52,66),(43,77,53,67),(44,78,54,68),(45,79,55,69),(46,80,56,70),(47,61,57,71),(48,62,58,72),(49,63,59,73),(50,64,60,74)], [(21,53,67),(22,54,68),(23,55,69),(24,56,70),(25,57,71),(26,58,72),(27,59,73),(28,60,74),(29,41,75),(30,42,76),(31,43,77),(32,44,78),(33,45,79),(34,46,80),(35,47,61),(36,48,62),(37,49,63),(38,50,64),(39,51,65),(40,52,66)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(36,40),(37,39),(41,73),(42,72),(43,71),(44,70),(45,69),(46,68),(47,67),(48,66),(49,65),(50,64),(51,63),(52,62),(53,61),(54,80),(55,79),(56,78),(57,77),(58,76),(59,75),(60,74)])
Matrix representation ►G ⊆ GL4(𝔽241) generated by
| 147 | 154 | 87 | 154 |
| 0 | 60 | 87 | 0 |
| 0 | 154 | 147 | 0 |
| 87 | 154 | 0 | 60 |
| 1 | 0 | 239 | 0 |
| 0 | 0 | 240 | 1 |
| 1 | 0 | 240 | 0 |
| 1 | 240 | 240 | 0 |
| 240 | 0 | 0 | 2 |
| 240 | 0 | 1 | 1 |
| 0 | 240 | 0 | 1 |
| 240 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 240 | 0 | 0 |
| 1 | 0 | 240 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [147,0,0,87,154,60,154,154,87,87,147,0,154,0,0,60],[1,0,1,1,0,0,0,240,239,240,240,240,0,1,0,0],[240,240,0,240,0,0,240,0,0,1,0,0,2,1,1,1],[1,0,1,1,0,0,240,0,0,0,0,240,0,1,0,0],[1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,240] >;
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 60 | 60 | 8 | 2 | 6 | 2 | 2 | 8 | 60 | 60 | 2 | 2 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
41 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D5 | D6 | D10 | D15 | D30 | S4 | C2×S4 | C4.3S4 | C20.3S4 | C5⋊S4 | C2×C5⋊S4 |
| kernel | C20.3S4 | Q8⋊D15 | C5×C4.A4 | C5×C4○D4 | C4.A4 | C5×Q8 | SL2(𝔽3) | C4○D4 | Q8 | C20 | C10 | C5 | C1 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 3 | 12 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_{20}._3S_4 % in TeX
G:=Group("C20.3S4"); // GroupNames label
G:=SmallGroup(480,1032);
// by ID
G=gap.SmallGroup(480,1032);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,3389,1688,170,1347,4204,3168,172,2525,1909,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^20=d^3=e^2=1,b^2=c^2=a^10,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,c*b*c^-1=a^10*b,d*b*d^-1=a^10*b*c,e*b*e=b*c,d*c*d^-1=b,e*c*e=a^10*c,e*d*e=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀