direct product, non-abelian, soluble
Aliases: C5×Q8.D6, GL2(𝔽3)⋊1C10, CSU2(𝔽3)⋊1C10, C2.7(C10×S4), (C2×C10).3S4, (Q8×C10)⋊4S3, C10.32(C2×S4), C22.2(C5×S4), Q8.2(S3×C10), (C5×Q8).14D6, (C5×GL2(𝔽3))⋊5C2, (C5×CSU2(𝔽3))⋊5C2, (C10×SL2(𝔽3))⋊8C2, (C2×SL2(𝔽3))⋊3C10, SL2(𝔽3).2(C2×C10), (C5×SL2(𝔽3)).14C22, (C2×Q8)⋊2(C5×S3), SmallGroup(480,1018)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C5×SL2(𝔽3) — C5×GL2(𝔽3) — C5×Q8.D6 |
| SL2(𝔽3) — C5×Q8.D6 |
Subgroups: 266 in 78 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×3], C22, C22, C5, S3, C6 [×2], C8 [×2], C2×C4 [×2], D4 [×2], Q8, Q8 [×2], C10, C10 [×2], Dic3, D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C20 [×3], C2×C10, C2×C10, SL2(𝔽3), C3⋊D4, C5×S3, C30 [×2], C8.C22, C40 [×2], C2×C20 [×2], C5×D4 [×2], C5×Q8, C5×Q8 [×2], CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), C5×Dic3, S3×C10, C2×C30, C5×M4(2), C5×SD16 [×2], C5×Q16 [×2], Q8×C10, C5×C4○D4, Q8.D6, C5×SL2(𝔽3), C5×C3⋊D4, C5×C8.C22, C5×CSU2(𝔽3), C5×GL2(𝔽3), C10×SL2(𝔽3), C5×Q8.D6
Quotients:
C1, C2 [×3], C22, C5, S3, C10 [×3], D6, C2×C10, S4, C5×S3, C2×S4, S3×C10, Q8.D6, C5×S4, C10×S4, C5×Q8.D6
Generators and relations
G = < a,b,c,d,e | a5=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, dbd-1=bc, dcd-1=b, ece-1=b-1c, ede-1=b2d-1 >
►On 80 points
Generators in S80
(1 16 11 19 9)(2 15 12 20 10)(3 13 17 8 5)(4 14 18 7 6)(21 67 45 72 58)(22 68 46 73 59)(23 63 47 74 60)(24 64 48 69 61)(25 65 49 70 62)(26 66 50 71 57)(27 77 41 33 53)(28 78 42 34 54)(29 79 43 35 55)(30 80 44 36 56)(31 75 39 37 51)(32 76 40 38 52)
(1 48 6 31)(2 45 5 28)(3 78 15 72)(4 75 16 69)(7 51 9 64)(8 54 10 67)(11 61 14 39)(12 58 13 42)(17 34 20 21)(18 37 19 24)(22 33 35 26)(23 25 36 38)(27 29 50 46)(30 32 47 49)(40 60 62 44)(41 43 57 59)(52 63 65 56)(53 55 66 68)(70 80 76 74)(71 73 77 79)
(1 46 6 29)(2 49 5 32)(3 76 15 70)(4 79 16 73)(7 55 9 68)(8 52 10 65)(11 59 14 43)(12 62 13 40)(17 38 20 25)(18 35 19 22)(21 23 34 36)(24 26 37 33)(27 48 50 31)(28 30 45 47)(39 41 61 57)(42 44 58 60)(51 53 64 66)(54 56 67 63)(69 71 75 77)(72 74 78 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 2 6 5)(3 16 15 4)(7 8 9 10)(11 12 14 13)(17 19 20 18)(21 24 34 37)(22 36 35 23)(25 26 38 33)(27 49 50 32)(28 31 45 48)(29 47 46 30)(39 58 61 42)(40 41 62 57)(43 60 59 44)(51 67 64 54)(52 53 65 66)(55 63 68 56)(69 78 75 72)(70 71 76 77)(73 80 79 74)
G:=sub<Sym(80)| (1,16,11,19,9)(2,15,12,20,10)(3,13,17,8,5)(4,14,18,7,6)(21,67,45,72,58)(22,68,46,73,59)(23,63,47,74,60)(24,64,48,69,61)(25,65,49,70,62)(26,66,50,71,57)(27,77,41,33,53)(28,78,42,34,54)(29,79,43,35,55)(30,80,44,36,56)(31,75,39,37,51)(32,76,40,38,52), (1,48,6,31)(2,45,5,28)(3,78,15,72)(4,75,16,69)(7,51,9,64)(8,54,10,67)(11,61,14,39)(12,58,13,42)(17,34,20,21)(18,37,19,24)(22,33,35,26)(23,25,36,38)(27,29,50,46)(30,32,47,49)(40,60,62,44)(41,43,57,59)(52,63,65,56)(53,55,66,68)(70,80,76,74)(71,73,77,79), (1,46,6,29)(2,49,5,32)(3,76,15,70)(4,79,16,73)(7,55,9,68)(8,52,10,65)(11,59,14,43)(12,62,13,40)(17,38,20,25)(18,35,19,22)(21,23,34,36)(24,26,37,33)(27,48,50,31)(28,30,45,47)(39,41,61,57)(42,44,58,60)(51,53,64,66)(54,56,67,63)(69,71,75,77)(72,74,78,80), (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), (1,2,6,5)(3,16,15,4)(7,8,9,10)(11,12,14,13)(17,19,20,18)(21,24,34,37)(22,36,35,23)(25,26,38,33)(27,49,50,32)(28,31,45,48)(29,47,46,30)(39,58,61,42)(40,41,62,57)(43,60,59,44)(51,67,64,54)(52,53,65,66)(55,63,68,56)(69,78,75,72)(70,71,76,77)(73,80,79,74)>;
G:=Group( (1,16,11,19,9)(2,15,12,20,10)(3,13,17,8,5)(4,14,18,7,6)(21,67,45,72,58)(22,68,46,73,59)(23,63,47,74,60)(24,64,48,69,61)(25,65,49,70,62)(26,66,50,71,57)(27,77,41,33,53)(28,78,42,34,54)(29,79,43,35,55)(30,80,44,36,56)(31,75,39,37,51)(32,76,40,38,52), (1,48,6,31)(2,45,5,28)(3,78,15,72)(4,75,16,69)(7,51,9,64)(8,54,10,67)(11,61,14,39)(12,58,13,42)(17,34,20,21)(18,37,19,24)(22,33,35,26)(23,25,36,38)(27,29,50,46)(30,32,47,49)(40,60,62,44)(41,43,57,59)(52,63,65,56)(53,55,66,68)(70,80,76,74)(71,73,77,79), (1,46,6,29)(2,49,5,32)(3,76,15,70)(4,79,16,73)(7,55,9,68)(8,52,10,65)(11,59,14,43)(12,62,13,40)(17,38,20,25)(18,35,19,22)(21,23,34,36)(24,26,37,33)(27,48,50,31)(28,30,45,47)(39,41,61,57)(42,44,58,60)(51,53,64,66)(54,56,67,63)(69,71,75,77)(72,74,78,80), (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), (1,2,6,5)(3,16,15,4)(7,8,9,10)(11,12,14,13)(17,19,20,18)(21,24,34,37)(22,36,35,23)(25,26,38,33)(27,49,50,32)(28,31,45,48)(29,47,46,30)(39,58,61,42)(40,41,62,57)(43,60,59,44)(51,67,64,54)(52,53,65,66)(55,63,68,56)(69,78,75,72)(70,71,76,77)(73,80,79,74) );
G=PermutationGroup([(1,16,11,19,9),(2,15,12,20,10),(3,13,17,8,5),(4,14,18,7,6),(21,67,45,72,58),(22,68,46,73,59),(23,63,47,74,60),(24,64,48,69,61),(25,65,49,70,62),(26,66,50,71,57),(27,77,41,33,53),(28,78,42,34,54),(29,79,43,35,55),(30,80,44,36,56),(31,75,39,37,51),(32,76,40,38,52)], [(1,48,6,31),(2,45,5,28),(3,78,15,72),(4,75,16,69),(7,51,9,64),(8,54,10,67),(11,61,14,39),(12,58,13,42),(17,34,20,21),(18,37,19,24),(22,33,35,26),(23,25,36,38),(27,29,50,46),(30,32,47,49),(40,60,62,44),(41,43,57,59),(52,63,65,56),(53,55,66,68),(70,80,76,74),(71,73,77,79)], [(1,46,6,29),(2,49,5,32),(3,76,15,70),(4,79,16,73),(7,55,9,68),(8,52,10,65),(11,59,14,43),(12,62,13,40),(17,38,20,25),(18,35,19,22),(21,23,34,36),(24,26,37,33),(27,48,50,31),(28,30,45,47),(39,41,61,57),(42,44,58,60),(51,53,64,66),(54,56,67,63),(69,71,75,77),(72,74,78,80)], [(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)], [(1,2,6,5),(3,16,15,4),(7,8,9,10),(11,12,14,13),(17,19,20,18),(21,24,34,37),(22,36,35,23),(25,26,38,33),(27,49,50,32),(28,31,45,48),(29,47,46,30),(39,58,61,42),(40,41,62,57),(43,60,59,44),(51,67,64,54),(52,53,65,66),(55,63,68,56),(69,78,75,72),(70,71,76,77),(73,80,79,74)])
Matrix representation ►G ⊆ GL4(𝔽241) generated by
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 |
| 0 | 171 | 70 | 171 |
| 70 | 171 | 0 | 70 |
| 171 | 171 | 171 | 0 |
| 70 | 0 | 171 | 171 |
| 0 | 70 | 171 | 70 |
| 70 | 70 | 70 | 0 |
| 171 | 70 | 0 | 171 |
| 70 | 0 | 171 | 171 |
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,98,0,0,0,0,98],[0,0,240,0,0,0,0,240,1,0,0,0,0,1,0,0],[0,1,0,0,240,0,0,0,0,0,0,240,0,0,1,0],[0,70,171,70,171,171,171,0,70,0,171,171,171,70,0,171],[0,70,171,70,70,70,70,0,171,70,0,171,70,0,171,171] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 30A | ··· | 30L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 12 | 8 | 6 | 6 | 12 | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 8 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 8 | ··· | 8 | 12 | ··· | 12 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D6 | C5×S3 | S3×C10 | S4 | C2×S4 | C5×S4 | C10×S4 | Q8.D6 | Q8.D6 | C5×Q8.D6 |
| kernel | C5×Q8.D6 | C5×CSU2(𝔽3) | C5×GL2(𝔽3) | C10×SL2(𝔽3) | Q8.D6 | CSU2(𝔽3) | GL2(𝔽3) | C2×SL2(𝔽3) | Q8×C10 | C5×Q8 | C2×Q8 | Q8 | C2×C10 | C10 | C22 | C2 | C5 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 | 1 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_5\times Q_8.D_6
% in TeX
G:=Group("C5xQ8.D6"); // GroupNames label
G:=SmallGroup(480,1018);
// by ID
G=gap.SmallGroup(480,1018);
# by ID
G:=PCGroup([7,-2,-2,-5,-3,-2,2,-2,3389,1123,4204,655,172,2525,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=d^6=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*b*e^-1=b^-1,d*b*d^-1=b*c,d*c*d^-1=b,e*c*e^-1=b^-1*c,e*d*e^-1=b^2*d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀