direct product, non-abelian, soluble
Aliases: C10×GL2(𝔽3), Q8⋊(S3×C10), (C5×Q8)⋊3D6, C2.6(C10×S4), (Q8×C10)⋊3S3, C10.31(C2×S4), (C2×C10).11S4, C22.5(C5×S4), (C10×SL2(𝔽3))⋊7C2, (C2×SL2(𝔽3))⋊2C10, SL2(𝔽3)⋊1(C2×C10), (C5×SL2(𝔽3))⋊9C22, (C2×Q8)⋊1(C5×S3), SmallGroup(480,1017)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C5×SL2(𝔽3) — C5×GL2(𝔽3) — C10×GL2(𝔽3) |
| SL2(𝔽3) — C10×GL2(𝔽3) |
Subgroups: 386 in 102 conjugacy classes, 24 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C22, C22 [×4], C5, S3 [×4], C6 [×3], C8 [×2], C2×C4, D4 [×3], Q8, Q8, C23, C10, C10 [×2], C10 [×2], D6 [×6], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, C20 [×2], C2×C10, C2×C10 [×4], SL2(𝔽3), C22×S3, C5×S3 [×4], C30 [×3], C2×SD16, C40 [×2], C2×C20, C5×D4 [×3], C5×Q8, C5×Q8, C22×C10, GL2(𝔽3) [×2], C2×SL2(𝔽3), S3×C10 [×6], C2×C30, C2×C40, C5×SD16 [×4], D4×C10, Q8×C10, C2×GL2(𝔽3), C5×SL2(𝔽3), S3×C2×C10, C10×SD16, C5×GL2(𝔽3) [×2], C10×SL2(𝔽3), C10×GL2(𝔽3)
Quotients:
C1, C2 [×3], C22, C5, S3, C10 [×3], D6, C2×C10, S4, C5×S3, GL2(𝔽3) [×2], C2×S4, S3×C10, C2×GL2(𝔽3), C5×S4, C5×GL2(𝔽3) [×2], C10×S4, C10×GL2(𝔽3)
Generators and relations
G = < a,b,c,d,e | a10=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece=b-1, dbd-1=bc, ebe=b2c, dcd-1=b, 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 18 40 72)(2 19 31 73)(3 20 32 74)(4 11 33 75)(5 12 34 76)(6 13 35 77)(7 14 36 78)(8 15 37 79)(9 16 38 80)(10 17 39 71)(21 59 47 65)(22 60 48 66)(23 51 49 67)(24 52 50 68)(25 53 41 69)(26 54 42 70)(27 55 43 61)(28 56 44 62)(29 57 45 63)(30 58 46 64)
(1 22 40 48)(2 23 31 49)(3 24 32 50)(4 25 33 41)(5 26 34 42)(6 27 35 43)(7 28 36 44)(8 29 37 45)(9 30 38 46)(10 21 39 47)(11 69 75 53)(12 70 76 54)(13 61 77 55)(14 62 78 56)(15 63 79 57)(16 64 80 58)(17 65 71 59)(18 66 72 60)(19 67 73 51)(20 68 74 52)
(11 25 69)(12 26 70)(13 27 61)(14 28 62)(15 29 63)(16 30 64)(17 21 65)(18 22 66)(19 23 67)(20 24 68)(41 53 75)(42 54 76)(43 55 77)(44 56 78)(45 57 79)(46 58 80)(47 59 71)(48 60 72)(49 51 73)(50 52 74)
(1 40)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 21)(18 22)(19 23)(20 24)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 71)(48 72)(49 73)(50 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), (1,18,40,72)(2,19,31,73)(3,20,32,74)(4,11,33,75)(5,12,34,76)(6,13,35,77)(7,14,36,78)(8,15,37,79)(9,16,38,80)(10,17,39,71)(21,59,47,65)(22,60,48,66)(23,51,49,67)(24,52,50,68)(25,53,41,69)(26,54,42,70)(27,55,43,61)(28,56,44,62)(29,57,45,63)(30,58,46,64), (1,22,40,48)(2,23,31,49)(3,24,32,50)(4,25,33,41)(5,26,34,42)(6,27,35,43)(7,28,36,44)(8,29,37,45)(9,30,38,46)(10,21,39,47)(11,69,75,53)(12,70,76,54)(13,61,77,55)(14,62,78,56)(15,63,79,57)(16,64,80,58)(17,65,71,59)(18,66,72,60)(19,67,73,51)(20,68,74,52), (11,25,69)(12,26,70)(13,27,61)(14,28,62)(15,29,63)(16,30,64)(17,21,65)(18,22,66)(19,23,67)(20,24,68)(41,53,75)(42,54,76)(43,55,77)(44,56,78)(45,57,79)(46,58,80)(47,59,71)(48,60,72)(49,51,73)(50,52,74), (1,40)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,21)(18,22)(19,23)(20,24)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,71)(48,72)(49,73)(50,74)>;
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), (1,18,40,72)(2,19,31,73)(3,20,32,74)(4,11,33,75)(5,12,34,76)(6,13,35,77)(7,14,36,78)(8,15,37,79)(9,16,38,80)(10,17,39,71)(21,59,47,65)(22,60,48,66)(23,51,49,67)(24,52,50,68)(25,53,41,69)(26,54,42,70)(27,55,43,61)(28,56,44,62)(29,57,45,63)(30,58,46,64), (1,22,40,48)(2,23,31,49)(3,24,32,50)(4,25,33,41)(5,26,34,42)(6,27,35,43)(7,28,36,44)(8,29,37,45)(9,30,38,46)(10,21,39,47)(11,69,75,53)(12,70,76,54)(13,61,77,55)(14,62,78,56)(15,63,79,57)(16,64,80,58)(17,65,71,59)(18,66,72,60)(19,67,73,51)(20,68,74,52), (11,25,69)(12,26,70)(13,27,61)(14,28,62)(15,29,63)(16,30,64)(17,21,65)(18,22,66)(19,23,67)(20,24,68)(41,53,75)(42,54,76)(43,55,77)(44,56,78)(45,57,79)(46,58,80)(47,59,71)(48,60,72)(49,51,73)(50,52,74), (1,40)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,21)(18,22)(19,23)(20,24)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,71)(48,72)(49,73)(50,74) );
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)], [(1,18,40,72),(2,19,31,73),(3,20,32,74),(4,11,33,75),(5,12,34,76),(6,13,35,77),(7,14,36,78),(8,15,37,79),(9,16,38,80),(10,17,39,71),(21,59,47,65),(22,60,48,66),(23,51,49,67),(24,52,50,68),(25,53,41,69),(26,54,42,70),(27,55,43,61),(28,56,44,62),(29,57,45,63),(30,58,46,64)], [(1,22,40,48),(2,23,31,49),(3,24,32,50),(4,25,33,41),(5,26,34,42),(6,27,35,43),(7,28,36,44),(8,29,37,45),(9,30,38,46),(10,21,39,47),(11,69,75,53),(12,70,76,54),(13,61,77,55),(14,62,78,56),(15,63,79,57),(16,64,80,58),(17,65,71,59),(18,66,72,60),(19,67,73,51),(20,68,74,52)], [(11,25,69),(12,26,70),(13,27,61),(14,28,62),(15,29,63),(16,30,64),(17,21,65),(18,22,66),(19,23,67),(20,24,68),(41,53,75),(42,54,76),(43,55,77),(44,56,78),(45,57,79),(46,58,80),(47,59,71),(48,60,72),(49,51,73),(50,52,74)], [(1,40),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,21),(18,22),(19,23),(20,24),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,71),(48,72),(49,73),(50,74)])
Matrix representation ►G ⊆ GL4(𝔽241) generated by
| 154 | 0 | 0 | 0 |
| 0 | 154 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 135 | 174 |
| 0 | 0 | 67 | 106 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 173 | 136 |
| 0 | 0 | 67 | 68 |
| 0 | 240 | 0 | 0 |
| 1 | 240 | 0 | 0 |
| 0 | 0 | 240 | 1 |
| 0 | 0 | 240 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [154,0,0,0,0,154,0,0,0,0,98,0,0,0,0,98],[1,0,0,0,0,1,0,0,0,0,135,67,0,0,174,106],[1,0,0,0,0,1,0,0,0,0,173,67,0,0,136,68],[0,1,0,0,240,240,0,0,0,0,240,240,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,240,240] >;
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 8 | 6 | 6 | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 12 | ··· | 12 | 8 | 8 | 8 | 8 | 6 | ··· | 6 | 8 | ··· | 8 | 6 | ··· | 6 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D6 | C5×S3 | GL2(𝔽3) | S3×C10 | C5×GL2(𝔽3) | S4 | C2×S4 | C5×S4 | C10×S4 | GL2(𝔽3) | C5×GL2(𝔽3) |
| kernel | C10×GL2(𝔽3) | C5×GL2(𝔽3) | C10×SL2(𝔽3) | C2×GL2(𝔽3) | GL2(𝔽3) | C2×SL2(𝔽3) | Q8×C10 | C5×Q8 | C2×Q8 | C10 | Q8 | C2 | C2×C10 | C10 | C22 | C2 | C10 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 1 | 1 | 4 | 4 | 4 | 16 | 2 | 2 | 8 | 8 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_{10}\times GL_2({\mathbb F}_3) % in TeX
G:=Group("C10xGL(2,3)"); // GroupNames label
G:=SmallGroup(480,1017);
// by ID
G=gap.SmallGroup(480,1017);
# by ID
G:=PCGroup([7,-2,-2,-5,-3,-2,2,-2,1123,4204,655,172,2525,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=d^3=e^2=1,c^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=b^-1,d*b*d^-1=b*c,e*b*e=b^2*c,d*c*d^-1=b,e*d*e=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀