metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: SD16⋊3F5, C5⋊C8.5D4, D4⋊D5⋊3C4, C5⋊2(C8○D8), (C8×F5)⋊5C2, C40⋊C2⋊4C4, C5⋊Q16⋊1C4, D4.F5⋊3C2, Q8.F5⋊1C2, D4.5(C2×F5), C2.21(D4×F5), C8.17(C2×F5), Q8.1(C2×F5), C40.16(C2×C4), D4⋊F5⋊3C2, Q8⋊2F5⋊1C2, (C5×SD16)⋊4C4, D20.3(C2×C4), C10.20(C4×D4), C40.C4⋊4C2, C4.7(C22×F5), C20.7(C22×C4), D10.3(C4○D4), C4.F5.3C22, D5⋊C8.12C22, Dic10.3(C2×C4), Dic5.74(C2×D4), (C4×D5).29C23, (C8×D5).28C22, (C4×F5).12C22, SD16⋊3D5.2C2, D4⋊2D5.6C22, Q8⋊2D5.4C22, C5⋊2C8.9(C2×C4), (C5×D4).5(C2×C4), (C5×Q8).1(C2×C4), SmallGroup(320,1074)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 402 in 106 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×3], C4, C4 [×5], C22 [×3], C5, C8, C8 [×5], C2×C4 [×4], D4, D4 [×3], Q8, Q8, D5 [×2], C10, C10, C42, C2×C8 [×4], M4(2) [×4], D8, SD16, SD16, Q16, C4○D4 [×2], Dic5, Dic5, C20, C20, F5 [×2], D10, D10, C2×C10, C4×C8, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C5⋊2C8, C40, C5⋊C8 [×2], C5⋊C8 [×2], Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C2×F5, C8○D8, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D5⋊C8, D5⋊C8, C4.F5 [×2], C4.F5, C4×F5, C2×C5⋊C8, C22.F5, D4⋊2D5, Q8⋊2D5, C8×F5, C40.C4, D4⋊F5, Q8⋊2F5, SD16⋊3D5, D4.F5, Q8.F5, SD16⋊3F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×F5 [×3], C8○D8, C22×F5, D4×F5, SD16⋊3F5
Generators and relations
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a3, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c3 >
►On 80 points
(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 74)(2 77)(3 80)(4 75)(5 78)(6 73)(7 76)(8 79)(9 28)(10 31)(11 26)(12 29)(13 32)(14 27)(15 30)(16 25)(17 40)(18 35)(19 38)(20 33)(21 36)(22 39)(23 34)(24 37)(41 62)(42 57)(43 60)(44 63)(45 58)(46 61)(47 64)(48 59)(49 69)(50 72)(51 67)(52 70)(53 65)(54 68)(55 71)(56 66)
(1 12 37 64 71)(2 13 38 57 72)(3 14 39 58 65)(4 15 40 59 66)(5 16 33 60 67)(6 9 34 61 68)(7 10 35 62 69)(8 11 36 63 70)(17 48 56 75 30)(18 41 49 76 31)(19 42 50 77 32)(20 43 51 78 25)(21 44 52 79 26)(22 45 53 80 27)(23 46 54 73 28)(24 47 55 74 29)
(9 34 68 61)(10 35 69 62)(11 36 70 63)(12 37 71 64)(13 38 72 57)(14 39 65 58)(15 40 66 59)(16 33 67 60)(17 50 44 28)(18 51 45 29)(19 52 46 30)(20 53 47 31)(21 54 48 32)(22 55 41 25)(23 56 42 26)(24 49 43 27)(73 75 77 79)(74 76 78 80)
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), (1,74)(2,77)(3,80)(4,75)(5,78)(6,73)(7,76)(8,79)(9,28)(10,31)(11,26)(12,29)(13,32)(14,27)(15,30)(16,25)(17,40)(18,35)(19,38)(20,33)(21,36)(22,39)(23,34)(24,37)(41,62)(42,57)(43,60)(44,63)(45,58)(46,61)(47,64)(48,59)(49,69)(50,72)(51,67)(52,70)(53,65)(54,68)(55,71)(56,66), (1,12,37,64,71)(2,13,38,57,72)(3,14,39,58,65)(4,15,40,59,66)(5,16,33,60,67)(6,9,34,61,68)(7,10,35,62,69)(8,11,36,63,70)(17,48,56,75,30)(18,41,49,76,31)(19,42,50,77,32)(20,43,51,78,25)(21,44,52,79,26)(22,45,53,80,27)(23,46,54,73,28)(24,47,55,74,29), (9,34,68,61)(10,35,69,62)(11,36,70,63)(12,37,71,64)(13,38,72,57)(14,39,65,58)(15,40,66,59)(16,33,67,60)(17,50,44,28)(18,51,45,29)(19,52,46,30)(20,53,47,31)(21,54,48,32)(22,55,41,25)(23,56,42,26)(24,49,43,27)(73,75,77,79)(74,76,78,80)>;
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), (1,74)(2,77)(3,80)(4,75)(5,78)(6,73)(7,76)(8,79)(9,28)(10,31)(11,26)(12,29)(13,32)(14,27)(15,30)(16,25)(17,40)(18,35)(19,38)(20,33)(21,36)(22,39)(23,34)(24,37)(41,62)(42,57)(43,60)(44,63)(45,58)(46,61)(47,64)(48,59)(49,69)(50,72)(51,67)(52,70)(53,65)(54,68)(55,71)(56,66), (1,12,37,64,71)(2,13,38,57,72)(3,14,39,58,65)(4,15,40,59,66)(5,16,33,60,67)(6,9,34,61,68)(7,10,35,62,69)(8,11,36,63,70)(17,48,56,75,30)(18,41,49,76,31)(19,42,50,77,32)(20,43,51,78,25)(21,44,52,79,26)(22,45,53,80,27)(23,46,54,73,28)(24,47,55,74,29), (9,34,68,61)(10,35,69,62)(11,36,70,63)(12,37,71,64)(13,38,72,57)(14,39,65,58)(15,40,66,59)(16,33,67,60)(17,50,44,28)(18,51,45,29)(19,52,46,30)(20,53,47,31)(21,54,48,32)(22,55,41,25)(23,56,42,26)(24,49,43,27)(73,75,77,79)(74,76,78,80) );
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)], [(1,74),(2,77),(3,80),(4,75),(5,78),(6,73),(7,76),(8,79),(9,28),(10,31),(11,26),(12,29),(13,32),(14,27),(15,30),(16,25),(17,40),(18,35),(19,38),(20,33),(21,36),(22,39),(23,34),(24,37),(41,62),(42,57),(43,60),(44,63),(45,58),(46,61),(47,64),(48,59),(49,69),(50,72),(51,67),(52,70),(53,65),(54,68),(55,71),(56,66)], [(1,12,37,64,71),(2,13,38,57,72),(3,14,39,58,65),(4,15,40,59,66),(5,16,33,60,67),(6,9,34,61,68),(7,10,35,62,69),(8,11,36,63,70),(17,48,56,75,30),(18,41,49,76,31),(19,42,50,77,32),(20,43,51,78,25),(21,44,52,79,26),(22,45,53,80,27),(23,46,54,73,28),(24,47,55,74,29)], [(9,34,68,61),(10,35,69,62),(11,36,70,63),(12,37,71,64),(13,38,72,57),(14,39,65,58),(15,40,66,59),(16,33,67,60),(17,50,44,28),(18,51,45,29),(19,52,46,30),(20,53,47,31),(21,54,48,32),(22,55,41,25),(23,56,42,26),(24,49,43,27),(73,75,77,79),(74,76,78,80)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 38 | 0 | 0 | 0 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [3,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,27,0,0,0,0,38,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[1,0,0,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 10A | 10B | 20A | 20B | 40A | 40B |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 20 | 20 | 40 | 40 |
| size | 1 | 1 | 4 | 10 | 20 | 2 | 4 | 5 | 5 | 10 | 10 | 10 | 10 | 20 | 4 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 4 | 16 | 8 | 16 | 8 | 8 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | C4○D4 | C8○D8 | F5 | C2×F5 | C2×F5 | C2×F5 | D4×F5 | SD16⋊3F5 |
| kernel | SD16⋊3F5 | C8×F5 | C40.C4 | D4⋊F5 | Q8⋊2F5 | SD16⋊3D5 | D4.F5 | Q8.F5 | C40⋊C2 | D4⋊D5 | C5⋊Q16 | C5×SD16 | C5⋊C8 | D10 | C5 | SD16 | C8 | D4 | Q8 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 1 | 2 |
In GAP, Magma, Sage, TeX
SD_{16}\rtimes_3F_5 % in TeX
G:=Group("SD16:3F5"); // GroupNames label
G:=SmallGroup(320,1074);
// by ID
G=gap.SmallGroup(320,1074);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,184,136,851,438,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=a^3,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations