?
G = SL2(𝔽3).11D10 order 480 = 25·3·5
1st non-split extension by SL2(𝔽3) of D10 acting through Inn(SL2(𝔽3))
Aliases: SL2(𝔽3).11D10, C5⋊D4.A4, Q8⋊2D5⋊C6, (Q8×D5)⋊1C6, C5⋊1(D4.A4), (Q8×C10)⋊2C6, Q8.2(C6×D5), D10.1(C2×A4), Q8.10D10⋊C3, C22.5(D5×A4), Dic5.A4⋊4C2, C10.6(C22×A4), Dic5.2(C2×A4), (C2×SL2(𝔽3))⋊1D5, (D5×SL2(𝔽3))⋊4C2, (C10×SL2(𝔽3))⋊6C2, (C5×SL2(𝔽3)).11C22, (C2×Q8)⋊(C3×D5), C2.7(C2×D5×A4), (C5×Q8).2(C2×C6), (C2×C10).13(C2×A4), SmallGroup(480,1040)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — C10 — C5×Q8 — C5×SL2(𝔽3) — D5×SL2(𝔽3) — SL2(𝔽3).11D10 |
| C5×Q8 — SL2(𝔽3).11D10 |
Subgroups: 574 in 92 conjugacy classes, 23 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×4], C22, C22 [×2], C5, C6 [×3], C2×C4 [×5], D4 [×4], Q8, Q8 [×3], D5 [×2], C10, C10, C12, C2×C6 [×2], C15, C2×Q8, C2×Q8 [×2], C4○D4 [×4], Dic5, Dic5, C20 [×2], D10, D10, C2×C10, SL2(𝔽3), C3×D4, C3×D5, C30 [×2], 2- (1+4), Dic10 [×2], C4×D5 [×4], D20 [×2], C5⋊D4, C5⋊D4, C2×C20, C5×Q8, C5×Q8, C2×SL2(𝔽3), C2×SL2(𝔽3), C4.A4, C3×Dic5, C6×D5, C2×C30, C4○D20 [×2], Q8×D5, Q8×D5, Q8⋊2D5, Q8⋊2D5, Q8×C10, D4.A4, C5×SL2(𝔽3), C3×C5⋊D4, Q8.10D10, Dic5.A4, D5×SL2(𝔽3), C10×SL2(𝔽3), SL2(𝔽3).11D10
Quotients:
C1, C2 [×3], C3, C22, C6 [×3], D5, A4, C2×C6, D10, C2×A4 [×3], C3×D5, C22×A4, C6×D5, D4.A4, D5×A4, C2×D5×A4, SL2(𝔽3).11D10
Generators and relations
G = < a,b,c,d,e | a4=c3=d10=1, b2=e2=a2, bab-1=a-1, cac-1=b, ad=da, ae=ea, cbc-1=ab, bd=db, be=eb, cd=dc, ce=ec, ede-1=a2d-1 >
►On 80 points
Generators in S80
(1 32 7 29)(2 33 8 30)(3 34 9 26)(4 35 10 27)(5 31 6 28)(11 16 23 40)(12 17 24 36)(13 18 25 37)(14 19 21 38)(15 20 22 39)(41 62 46 67)(42 63 47 68)(43 64 48 69)(44 65 49 70)(45 66 50 61)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(1 22 7 15)(2 23 8 11)(3 24 9 12)(4 25 10 13)(5 21 6 14)(16 30 40 33)(17 26 36 34)(18 27 37 35)(19 28 38 31)(20 29 39 32)(41 52 46 57)(42 53 47 58)(43 54 48 59)(44 55 49 60)(45 56 50 51)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)
(11 30 40)(12 26 36)(13 27 37)(14 28 38)(15 29 39)(16 23 33)(17 24 34)(18 25 35)(19 21 31)(20 22 32)(51 61 71)(52 62 72)(53 63 73)(54 64 74)(55 65 75)(56 66 76)(57 67 77)(58 68 78)(59 69 79)(60 70 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 50 7 45)(2 44 8 49)(3 48 9 43)(4 42 10 47)(5 46 6 41)(11 60 23 55)(12 54 24 59)(13 58 25 53)(14 52 21 57)(15 56 22 51)(16 75 40 80)(17 79 36 74)(18 73 37 78)(19 77 38 72)(20 71 39 76)(26 64 34 69)(27 68 35 63)(28 62 31 67)(29 66 32 61)(30 70 33 65)
G:=sub<Sym(80)| (1,32,7,29)(2,33,8,30)(3,34,9,26)(4,35,10,27)(5,31,6,28)(11,16,23,40)(12,17,24,36)(13,18,25,37)(14,19,21,38)(15,20,22,39)(41,62,46,67)(42,63,47,68)(43,64,48,69)(44,65,49,70)(45,66,50,61)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (1,22,7,15)(2,23,8,11)(3,24,9,12)(4,25,10,13)(5,21,6,14)(16,30,40,33)(17,26,36,34)(18,27,37,35)(19,28,38,31)(20,29,39,32)(41,52,46,57)(42,53,47,58)(43,54,48,59)(44,55,49,60)(45,56,50,51)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (11,30,40)(12,26,36)(13,27,37)(14,28,38)(15,29,39)(16,23,33)(17,24,34)(18,25,35)(19,21,31)(20,22,32)(51,61,71)(52,62,72)(53,63,73)(54,64,74)(55,65,75)(56,66,76)(57,67,77)(58,68,78)(59,69,79)(60,70,80), (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), (1,50,7,45)(2,44,8,49)(3,48,9,43)(4,42,10,47)(5,46,6,41)(11,60,23,55)(12,54,24,59)(13,58,25,53)(14,52,21,57)(15,56,22,51)(16,75,40,80)(17,79,36,74)(18,73,37,78)(19,77,38,72)(20,71,39,76)(26,64,34,69)(27,68,35,63)(28,62,31,67)(29,66,32,61)(30,70,33,65)>;
G:=Group( (1,32,7,29)(2,33,8,30)(3,34,9,26)(4,35,10,27)(5,31,6,28)(11,16,23,40)(12,17,24,36)(13,18,25,37)(14,19,21,38)(15,20,22,39)(41,62,46,67)(42,63,47,68)(43,64,48,69)(44,65,49,70)(45,66,50,61)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (1,22,7,15)(2,23,8,11)(3,24,9,12)(4,25,10,13)(5,21,6,14)(16,30,40,33)(17,26,36,34)(18,27,37,35)(19,28,38,31)(20,29,39,32)(41,52,46,57)(42,53,47,58)(43,54,48,59)(44,55,49,60)(45,56,50,51)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (11,30,40)(12,26,36)(13,27,37)(14,28,38)(15,29,39)(16,23,33)(17,24,34)(18,25,35)(19,21,31)(20,22,32)(51,61,71)(52,62,72)(53,63,73)(54,64,74)(55,65,75)(56,66,76)(57,67,77)(58,68,78)(59,69,79)(60,70,80), (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), (1,50,7,45)(2,44,8,49)(3,48,9,43)(4,42,10,47)(5,46,6,41)(11,60,23,55)(12,54,24,59)(13,58,25,53)(14,52,21,57)(15,56,22,51)(16,75,40,80)(17,79,36,74)(18,73,37,78)(19,77,38,72)(20,71,39,76)(26,64,34,69)(27,68,35,63)(28,62,31,67)(29,66,32,61)(30,70,33,65) );
G=PermutationGroup([(1,32,7,29),(2,33,8,30),(3,34,9,26),(4,35,10,27),(5,31,6,28),(11,16,23,40),(12,17,24,36),(13,18,25,37),(14,19,21,38),(15,20,22,39),(41,62,46,67),(42,63,47,68),(43,64,48,69),(44,65,49,70),(45,66,50,61),(51,76,56,71),(52,77,57,72),(53,78,58,73),(54,79,59,74),(55,80,60,75)], [(1,22,7,15),(2,23,8,11),(3,24,9,12),(4,25,10,13),(5,21,6,14),(16,30,40,33),(17,26,36,34),(18,27,37,35),(19,28,38,31),(20,29,39,32),(41,52,46,57),(42,53,47,58),(43,54,48,59),(44,55,49,60),(45,56,50,51),(61,71,66,76),(62,72,67,77),(63,73,68,78),(64,74,69,79),(65,75,70,80)], [(11,30,40),(12,26,36),(13,27,37),(14,28,38),(15,29,39),(16,23,33),(17,24,34),(18,25,35),(19,21,31),(20,22,32),(51,61,71),(52,62,72),(53,63,73),(54,64,74),(55,65,75),(56,66,76),(57,67,77),(58,68,78),(59,69,79),(60,70,80)], [(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)], [(1,50,7,45),(2,44,8,49),(3,48,9,43),(4,42,10,47),(5,46,6,41),(11,60,23,55),(12,54,24,59),(13,58,25,53),(14,52,21,57),(15,56,22,51),(16,75,40,80),(17,79,36,74),(18,73,37,78),(19,77,38,72),(20,71,39,76),(26,64,34,69),(27,68,35,63),(28,62,31,67),(29,66,32,61),(30,70,33,65)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 14 | 13 | 0 | 0 |
| 13 | 47 | 0 | 0 |
| 0 | 0 | 14 | 13 |
| 0 | 0 | 13 | 47 |
| 0 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 0 |
| 48 | 14 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 48 | 14 |
| 0 | 0 | 1 | 0 |
| 9 | 0 | 13 | 0 |
| 0 | 9 | 0 | 13 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 13 | 0 | 38 | 0 |
| 0 | 13 | 0 | 38 |
| 18 | 0 | 48 | 0 |
| 0 | 18 | 0 | 48 |
G:=sub<GL(4,GF(61))| [14,13,0,0,13,47,0,0,0,0,14,13,0,0,13,47],[0,1,0,0,60,0,0,0,0,0,0,1,0,0,60,0],[48,1,0,0,14,0,0,0,0,0,48,1,0,0,14,0],[9,0,0,0,0,9,0,0,13,0,27,0,0,13,0,27],[13,0,18,0,0,13,0,18,38,0,48,0,0,38,0,48] >;
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 10A | ··· | 10F | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 10 | 30 | 4 | 4 | 6 | 6 | 10 | 30 | 2 | 2 | 4 | 4 | 8 | 8 | 40 | 40 | 2 | ··· | 2 | 40 | 40 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D5 | D10 | C3×D5 | C6×D5 | A4 | C2×A4 | C2×A4 | C2×A4 | D4.A4 | D4.A4 | SL2(𝔽3).11D10 | D5×A4 | C2×D5×A4 |
| kernel | SL2(𝔽3).11D10 | Dic5.A4 | D5×SL2(𝔽3) | C10×SL2(𝔽3) | Q8.10D10 | Q8×D5 | Q8⋊2D5 | Q8×C10 | C2×SL2(𝔽3) | SL2(𝔽3) | C2×Q8 | Q8 | C5⋊D4 | Dic5 | D10 | C2×C10 | C5 | C5 | C1 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 2 | 12 | 2 | 2 |
In GAP, Magma, Sage, TeX
SL_2({\mathbb F}_3)._{11}D_{10} % in TeX
G:=Group("SL(2,3).11D10"); // GroupNames label
G:=SmallGroup(480,1040);
// by ID
G=gap.SmallGroup(480,1040);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,2,-5,-2,3389,269,584,123,795,382,8069]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^3=d^10=1,b^2=e^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,a*d=d*a,a*e=e*a,c*b*c^-1=a*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^2*d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀