Aliases: C20.6S4, Q8.4D30, SL2(𝔽3).10D10, C4.6(C5⋊S4), C4.A4⋊2D5, C4○D4⋊1D15, Q8⋊D15⋊7C2, C10.23(C2×S4), C5⋊3(C4.6S4), Q8.D15⋊7C2, (C5×Q8).11D6, (C5×SL2(𝔽3)).10C22, C2.9(C2×C5⋊S4), (C5×C4.A4)⋊2C2, (C5×C4○D4)⋊1S3, SmallGroup(480,1031)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×SL2(𝔽3) — C20.6S4 |
| C5×SL2(𝔽3) — C20.6S4 |
Subgroups: 626 in 78 conjugacy classes, 17 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×2], C22 [×2], C5, S3 [×2], C6, C8 [×2], C2×C4 [×2], D4 [×3], Q8, Q8, D5, C10, C10, Dic3, C12, D6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, SL2(𝔽3), C4×S3, D15 [×2], C30, C4○D8, C5⋊2C8 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×Q8, CSU2(𝔽3), GL2(𝔽3), C4.A4, Dic15, C60, D30, C2×C5⋊2C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, C4.6S4, C5×SL2(𝔽3), C4×D15, D4.8D10, Q8.D15, Q8⋊D15, C5×C4.A4, C20.6S4
Quotients:
C1, C2 [×3], C22, S3, D5, D6, D10, S4, D15, C2×S4, D30, C4.6S4, C5⋊S4, C2×C5⋊S4, C20.6S4
Generators and relations
G = < a,b,c,d,e | a20=d3=e2=1, b2=c2=a10, ab=ba, ac=ca, ad=da, eae=a9, 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 54 11 44)(2 55 12 45)(3 56 13 46)(4 57 14 47)(5 58 15 48)(6 59 16 49)(7 60 17 50)(8 41 18 51)(9 42 19 52)(10 43 20 53)(21 75 31 65)(22 76 32 66)(23 77 33 67)(24 78 34 68)(25 79 35 69)(26 80 36 70)(27 61 37 71)(28 62 38 72)(29 63 39 73)(30 64 40 74)
(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 60 31 50)(22 41 32 51)(23 42 33 52)(24 43 34 53)(25 44 35 54)(26 45 36 55)(27 46 37 56)(28 47 38 57)(29 48 39 58)(30 49 40 59)
(21 60 65)(22 41 66)(23 42 67)(24 43 68)(25 44 69)(26 45 70)(27 46 71)(28 47 72)(29 48 73)(30 49 74)(31 50 75)(32 51 76)(33 52 77)(34 53 78)(35 54 79)(36 55 80)(37 56 61)(38 57 62)(39 58 63)(40 59 64)
(1 11)(2 20)(3 9)(4 18)(5 7)(6 16)(8 14)(10 12)(13 19)(15 17)(21 48)(22 57)(23 46)(24 55)(25 44)(26 53)(27 42)(28 51)(29 60)(30 49)(31 58)(32 47)(33 56)(34 45)(35 54)(36 43)(37 52)(38 41)(39 50)(40 59)(61 77)(62 66)(63 75)(65 73)(67 71)(68 80)(70 78)(72 76)
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), (1,54,11,44)(2,55,12,45)(3,56,13,46)(4,57,14,47)(5,58,15,48)(6,59,16,49)(7,60,17,50)(8,41,18,51)(9,42,19,52)(10,43,20,53)(21,75,31,65)(22,76,32,66)(23,77,33,67)(24,78,34,68)(25,79,35,69)(26,80,36,70)(27,61,37,71)(28,62,38,72)(29,63,39,73)(30,64,40,74), (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)(21,60,31,50)(22,41,32,51)(23,42,33,52)(24,43,34,53)(25,44,35,54)(26,45,36,55)(27,46,37,56)(28,47,38,57)(29,48,39,58)(30,49,40,59), (21,60,65)(22,41,66)(23,42,67)(24,43,68)(25,44,69)(26,45,70)(27,46,71)(28,47,72)(29,48,73)(30,49,74)(31,50,75)(32,51,76)(33,52,77)(34,53,78)(35,54,79)(36,55,80)(37,56,61)(38,57,62)(39,58,63)(40,59,64), (1,11)(2,20)(3,9)(4,18)(5,7)(6,16)(8,14)(10,12)(13,19)(15,17)(21,48)(22,57)(23,46)(24,55)(25,44)(26,53)(27,42)(28,51)(29,60)(30,49)(31,58)(32,47)(33,56)(34,45)(35,54)(36,43)(37,52)(38,41)(39,50)(40,59)(61,77)(62,66)(63,75)(65,73)(67,71)(68,80)(70,78)(72,76)>;
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), (1,54,11,44)(2,55,12,45)(3,56,13,46)(4,57,14,47)(5,58,15,48)(6,59,16,49)(7,60,17,50)(8,41,18,51)(9,42,19,52)(10,43,20,53)(21,75,31,65)(22,76,32,66)(23,77,33,67)(24,78,34,68)(25,79,35,69)(26,80,36,70)(27,61,37,71)(28,62,38,72)(29,63,39,73)(30,64,40,74), (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)(21,60,31,50)(22,41,32,51)(23,42,33,52)(24,43,34,53)(25,44,35,54)(26,45,36,55)(27,46,37,56)(28,47,38,57)(29,48,39,58)(30,49,40,59), (21,60,65)(22,41,66)(23,42,67)(24,43,68)(25,44,69)(26,45,70)(27,46,71)(28,47,72)(29,48,73)(30,49,74)(31,50,75)(32,51,76)(33,52,77)(34,53,78)(35,54,79)(36,55,80)(37,56,61)(38,57,62)(39,58,63)(40,59,64), (1,11)(2,20)(3,9)(4,18)(5,7)(6,16)(8,14)(10,12)(13,19)(15,17)(21,48)(22,57)(23,46)(24,55)(25,44)(26,53)(27,42)(28,51)(29,60)(30,49)(31,58)(32,47)(33,56)(34,45)(35,54)(36,43)(37,52)(38,41)(39,50)(40,59)(61,77)(62,66)(63,75)(65,73)(67,71)(68,80)(70,78)(72,76) );
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)], [(1,54,11,44),(2,55,12,45),(3,56,13,46),(4,57,14,47),(5,58,15,48),(6,59,16,49),(7,60,17,50),(8,41,18,51),(9,42,19,52),(10,43,20,53),(21,75,31,65),(22,76,32,66),(23,77,33,67),(24,78,34,68),(25,79,35,69),(26,80,36,70),(27,61,37,71),(28,62,38,72),(29,63,39,73),(30,64,40,74)], [(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),(21,60,31,50),(22,41,32,51),(23,42,33,52),(24,43,34,53),(25,44,35,54),(26,45,36,55),(27,46,37,56),(28,47,38,57),(29,48,39,58),(30,49,40,59)], [(21,60,65),(22,41,66),(23,42,67),(24,43,68),(25,44,69),(26,45,70),(27,46,71),(28,47,72),(29,48,73),(30,49,74),(31,50,75),(32,51,76),(33,52,77),(34,53,78),(35,54,79),(36,55,80),(37,56,61),(38,57,62),(39,58,63),(40,59,64)], [(1,11),(2,20),(3,9),(4,18),(5,7),(6,16),(8,14),(10,12),(13,19),(15,17),(21,48),(22,57),(23,46),(24,55),(25,44),(26,53),(27,42),(28,51),(29,60),(30,49),(31,58),(32,47),(33,56),(34,45),(35,54),(36,43),(37,52),(38,41),(39,50),(40,59),(61,77),(62,66),(63,75),(65,73),(67,71),(68,80),(70,78),(72,76)])
Matrix representation ►G ⊆ GL4(𝔽241) generated by
| 219 | 233 | 0 | 0 |
| 8 | 211 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 177 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 12 |
| 0 | 0 | 26 | 228 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 216 | 12 |
| 0 | 0 | 229 | 25 |
| 0 | 240 | 0 | 0 |
| 1 | 240 | 0 | 0 |
| 0 | 0 | 240 | 1 |
| 0 | 0 | 240 | 0 |
| 1 | 240 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 240 | 0 |
G:=sub<GL(4,GF(241))| [219,8,0,0,233,211,0,0,0,0,177,0,0,0,0,177],[1,0,0,0,0,1,0,0,0,0,13,26,0,0,12,228],[1,0,0,0,0,1,0,0,0,0,216,229,0,0,12,25],[0,1,0,0,240,240,0,0,0,0,240,240,0,0,1,0],[1,0,0,0,240,240,0,0,0,0,0,240,0,0,240,0] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 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 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 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 | 8 | 1 | 1 | 6 | 60 | 2 | 2 | 8 | 30 | 30 | 30 | 30 | 2 | 2 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D5 | D6 | D10 | D15 | D30 | C4.6S4 | S4 | C2×S4 | C4.6S4 | C20.6S4 | C5⋊S4 | C2×C5⋊S4 |
| kernel | C20.6S4 | Q8.D15 | Q8⋊D15 | C5×C4.A4 | C5×C4○D4 | C4.A4 | C5×Q8 | SL2(𝔽3) | C4○D4 | Q8 | C5 | C20 | C10 | C5 | C1 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 12 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_{20}._6S_4 % in TeX
G:=Group("C20.6S4"); // GroupNames label
G:=SmallGroup(480,1031);
// by ID
G=gap.SmallGroup(480,1031);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1688,170,1347,4204,3168,172,2525,1909,285,124]);
// 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^9,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
○
≀