direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C8⋊D5, C8⋊9D10, C40⋊11C22, C10⋊3M4(2), C20.36C23, (C2×C8)⋊6D5, (C2×C40)⋊9C2, (C4×D5).3C4, C4.24(C4×D5), C5⋊4(C2×M4(2)), C20.48(C2×C4), (C2×C4).98D10, C5⋊2C8⋊10C22, D10.10(C2×C4), (C2×Dic5).6C4, (C22×D5).4C4, C22.14(C4×D5), C4.36(C22×D5), C10.26(C22×C4), Dic5.12(C2×C4), (C4×D5).22C22, (C2×C20).111C22, C2.14(C2×C4×D5), (C2×C4×D5).12C2, (C2×C5⋊2C8)⋊11C2, (C2×C10).35(C2×C4), SmallGroup(160,121)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C8⋊D5
G = < a,b,c,d | a2=b8=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Subgroups: 184 in 68 conjugacy classes, 41 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C2×M4(2), C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C8⋊D5, C2×C5⋊2C8, C2×C40, C2×C4×D5, C2×C8⋊D5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, M4(2), C22×C4, D10, C2×M4(2), C4×D5, C22×D5, C8⋊D5, C2×C4×D5, C2×C8⋊D5
►On 80 points
(1 72)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 38)(18 39)(19 40)(20 33)(21 34)(22 35)(23 36)(24 37)(41 61)(42 62)(43 63)(44 64)(45 57)(46 58)(47 59)(48 60)(49 77)(50 78)(51 79)(52 80)(53 73)(54 74)(55 75)(56 76)
(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 14 37 62 75)(2 15 38 63 76)(3 16 39 64 77)(4 9 40 57 78)(5 10 33 58 79)(6 11 34 59 80)(7 12 35 60 73)(8 13 36 61 74)(17 43 56 65 30)(18 44 49 66 31)(19 45 50 67 32)(20 46 51 68 25)(21 47 52 69 26)(22 48 53 70 27)(23 41 54 71 28)(24 42 55 72 29)
(1 75)(2 80)(3 77)(4 74)(5 79)(6 76)(7 73)(8 78)(9 61)(10 58)(11 63)(12 60)(13 57)(14 62)(15 59)(16 64)(17 21)(19 23)(25 46)(26 43)(27 48)(28 45)(29 42)(30 47)(31 44)(32 41)(34 38)(36 40)(49 66)(50 71)(51 68)(52 65)(53 70)(54 67)(55 72)(56 69)
G:=sub<Sym(80)| (1,72)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37)(41,61)(42,62)(43,63)(44,64)(45,57)(46,58)(47,59)(48,60)(49,77)(50,78)(51,79)(52,80)(53,73)(54,74)(55,75)(56,76), (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), (1,14,37,62,75)(2,15,38,63,76)(3,16,39,64,77)(4,9,40,57,78)(5,10,33,58,79)(6,11,34,59,80)(7,12,35,60,73)(8,13,36,61,74)(17,43,56,65,30)(18,44,49,66,31)(19,45,50,67,32)(20,46,51,68,25)(21,47,52,69,26)(22,48,53,70,27)(23,41,54,71,28)(24,42,55,72,29), (1,75)(2,80)(3,77)(4,74)(5,79)(6,76)(7,73)(8,78)(9,61)(10,58)(11,63)(12,60)(13,57)(14,62)(15,59)(16,64)(17,21)(19,23)(25,46)(26,43)(27,48)(28,45)(29,42)(30,47)(31,44)(32,41)(34,38)(36,40)(49,66)(50,71)(51,68)(52,65)(53,70)(54,67)(55,72)(56,69)>;
G:=Group( (1,72)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37)(41,61)(42,62)(43,63)(44,64)(45,57)(46,58)(47,59)(48,60)(49,77)(50,78)(51,79)(52,80)(53,73)(54,74)(55,75)(56,76), (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), (1,14,37,62,75)(2,15,38,63,76)(3,16,39,64,77)(4,9,40,57,78)(5,10,33,58,79)(6,11,34,59,80)(7,12,35,60,73)(8,13,36,61,74)(17,43,56,65,30)(18,44,49,66,31)(19,45,50,67,32)(20,46,51,68,25)(21,47,52,69,26)(22,48,53,70,27)(23,41,54,71,28)(24,42,55,72,29), (1,75)(2,80)(3,77)(4,74)(5,79)(6,76)(7,73)(8,78)(9,61)(10,58)(11,63)(12,60)(13,57)(14,62)(15,59)(16,64)(17,21)(19,23)(25,46)(26,43)(27,48)(28,45)(29,42)(30,47)(31,44)(32,41)(34,38)(36,40)(49,66)(50,71)(51,68)(52,65)(53,70)(54,67)(55,72)(56,69) );
G=PermutationGroup([[(1,72),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,32),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,38),(18,39),(19,40),(20,33),(21,34),(22,35),(23,36),(24,37),(41,61),(42,62),(43,63),(44,64),(45,57),(46,58),(47,59),(48,60),(49,77),(50,78),(51,79),(52,80),(53,73),(54,74),(55,75),(56,76)], [(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)], [(1,14,37,62,75),(2,15,38,63,76),(3,16,39,64,77),(4,9,40,57,78),(5,10,33,58,79),(6,11,34,59,80),(7,12,35,60,73),(8,13,36,61,74),(17,43,56,65,30),(18,44,49,66,31),(19,45,50,67,32),(20,46,51,68,25),(21,47,52,69,26),(22,48,53,70,27),(23,41,54,71,28),(24,42,55,72,29)], [(1,75),(2,80),(3,77),(4,74),(5,79),(6,76),(7,73),(8,78),(9,61),(10,58),(11,63),(12,60),(13,57),(14,62),(15,59),(16,64),(17,21),(19,23),(25,46),(26,43),(27,48),(28,45),(29,42),(30,47),(31,44),(32,41),(34,38),(36,40),(49,66),(50,71),(51,68),(52,65),(53,70),(54,67),(55,72),(56,69)]])
C2×C8⋊D5 is a maximal subgroup of
C8.25D20 C20.10M4(2) (C2×C8)⋊F5 C20.24C42 C20.25C42 C8⋊6D20 D10.5C42 C8⋊9D20 D10.6C42 D10.7C42 D10⋊7M4(2) C22⋊C8⋊D5 Dic5⋊2M4(2) C5⋊2C8⋊26D4 (D4×D5)⋊C4 D4⋊(C4×D5) C5⋊2C8⋊D4 C5⋊(C8⋊2D4) (Q8×D5)⋊C4 Q8⋊(C4×D5) C5⋊(C8⋊D4) C5⋊2C8.D4 D10⋊5M4(2) C20⋊5M4(2) C20⋊6M4(2) C42.30D10 C8⋊(C4×D5) C8⋊2D20 C8.2D20 C40⋊20(C2×C4) C8⋊3D20 M4(2).25D10 C40⋊32D4 C40⋊18D4 C40⋊12D4 C40⋊8D4 C40.36D4 C2×D5×M4(2) C20.72C24 Q16⋊D10
C2×C8⋊D5 is a maximal quotient of
C40⋊11Q8 C42.282D10 C8⋊6D20 Dic5.9M4(2) D10⋊7M4(2) C5⋊2C8⋊26D4 C42.198D10 C42.202D10 C20⋊5M4(2) C20⋊6M4(2) C40⋊32D4
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | ··· | 20H | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D5 | M4(2) | D10 | D10 | C4×D5 | C4×D5 | C8⋊D5 |
| kernel | C2×C8⋊D5 | C8⋊D5 | C2×C5⋊2C8 | C2×C40 | C2×C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | C2×C8 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 16 |
Matrix representation of C2×C8⋊D5 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 1 | 0 | 0 |
| 0 | 6 | 2 |
| 0 | 39 | 35 |
| 1 | 0 | 0 |
| 0 | 0 | 40 |
| 0 | 1 | 6 |
| 40 | 0 | 0 |
| 0 | 6 | 35 |
| 0 | 40 | 35 |
G:=sub<GL(3,GF(41))| [40,0,0,0,40,0,0,0,40],[1,0,0,0,6,39,0,2,35],[1,0,0,0,0,1,0,40,6],[40,0,0,0,6,40,0,35,35] >;
C2×C8⋊D5 in GAP, Magma, Sage, TeX
C_2\times C_8\rtimes D_5
% in TeX
G:=Group("C2xC8:D5"); // GroupNames label
G:=SmallGroup(160,121);
// by ID
G=gap.SmallGroup(160,121);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,362,50,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations