direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D5.D8, D10.11D8, D10.6Q16, C8⋊7(C2×F5), (C2×C8)⋊5F5, (C2×C40)⋊5C4, C40⋊7(C2×C4), (C8×D5)⋊6C4, C10⋊(C2.D8), D5⋊(C2.D8), D5.1(C2×D8), (C4×D5).83D4, C4.13(C4⋊F5), C20.20(C4⋊C4), D5.1(C2×Q16), (C4×D5).25Q8, D10.30(C2×D4), C4⋊F5.16C22, D10.28(C4⋊C4), C4.36(C22×F5), C20.76(C22×C4), (C2×Dic5).34Q8, Dic5.14(C2×Q8), (C4×D5).76C23, (C22×D5).98D4, (C8×D5).53C22, C22.23(C4⋊F5), Dic5.29(C4⋊C4), C5⋊(C2×C2.D8), (D5×C2×C8).20C2, (C2×C5⋊2C8)⋊18C4, C2.15(C2×C4⋊F5), C5⋊2C8⋊35(C2×C4), C10.12(C2×C4⋊C4), (C2×C4⋊F5).14C2, (C4×D5).85(C2×C4), (C2×C4).137(C2×F5), (C2×C10).20(C4⋊C4), (C2×C20).127(C2×C4), (C2×C4×D5).395C22, SmallGroup(320,1058)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D5.D8
G = < a,b,c,d,e | a2=b5=c2=d8=1, e2=b-1c, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b3, cd=dc, ece-1=b2c, ede-1=d-1 >
Subgroups: 538 in 130 conjugacy classes, 60 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2.D8, C2×C4⋊C4, C22×C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C2×C2.D8, C8×D5, C2×C5⋊2C8, C2×C40, C4⋊F5, C4⋊F5, C2×C4×D5, C22×F5, D5.D8, D5×C2×C8, C2×C4⋊F5, C2×D5.D8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, F5, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, C2×F5, C2×C2.D8, C4⋊F5, C22×F5, D5.D8, C2×C4⋊F5, C2×D5.D8
►On 80 points
Generators in S80
(1 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 73)(32 74)(49 72)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)
(1 75 39 12 66)(2 76 40 13 67)(3 77 33 14 68)(4 78 34 15 69)(5 79 35 16 70)(6 80 36 9 71)(7 73 37 10 72)(8 74 38 11 65)(17 43 52 64 26)(18 44 53 57 27)(19 45 54 58 28)(20 46 55 59 29)(21 47 56 60 30)(22 48 49 61 31)(23 41 50 62 32)(24 42 51 63 25)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 65)(9 80)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)
(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)
(2 8)(3 7)(4 6)(9 78 36 69)(10 77 37 68)(11 76 38 67)(12 75 39 66)(13 74 40 65)(14 73 33 72)(15 80 34 71)(16 79 35 70)(17 50 43 32)(18 49 44 31)(19 56 45 30)(20 55 46 29)(21 54 47 28)(22 53 48 27)(23 52 41 26)(24 51 42 25)(57 61)(58 60)(62 64)
G:=sub<Sym(80)| (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(49,72)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71), (1,75,39,12,66)(2,76,40,13,67)(3,77,33,14,68)(4,78,34,15,69)(5,79,35,16,70)(6,80,36,9,71)(7,73,37,10,72)(8,74,38,11,65)(17,43,52,64,26)(18,44,53,57,27)(19,45,54,58,28)(20,46,55,59,29)(21,47,56,60,30)(22,48,49,61,31)(23,41,50,62,32)(24,42,51,63,25), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,65)(9,80)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60), (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), (2,8)(3,7)(4,6)(9,78,36,69)(10,77,37,68)(11,76,38,67)(12,75,39,66)(13,74,40,65)(14,73,33,72)(15,80,34,71)(16,79,35,70)(17,50,43,32)(18,49,44,31)(19,56,45,30)(20,55,46,29)(21,54,47,28)(22,53,48,27)(23,52,41,26)(24,51,42,25)(57,61)(58,60)(62,64)>;
G:=Group( (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(49,72)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71), (1,75,39,12,66)(2,76,40,13,67)(3,77,33,14,68)(4,78,34,15,69)(5,79,35,16,70)(6,80,36,9,71)(7,73,37,10,72)(8,74,38,11,65)(17,43,52,64,26)(18,44,53,57,27)(19,45,54,58,28)(20,46,55,59,29)(21,47,56,60,30)(22,48,49,61,31)(23,41,50,62,32)(24,42,51,63,25), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,65)(9,80)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60), (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), (2,8)(3,7)(4,6)(9,78,36,69)(10,77,37,68)(11,76,38,67)(12,75,39,66)(13,74,40,65)(14,73,33,72)(15,80,34,71)(16,79,35,70)(17,50,43,32)(18,49,44,31)(19,56,45,30)(20,55,46,29)(21,54,47,28)(22,53,48,27)(23,52,41,26)(24,51,42,25)(57,61)(58,60)(62,64) );
G=PermutationGroup([[(1,63),(2,64),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,47),(10,48),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,40),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,73),(32,74),(49,72),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71)], [(1,75,39,12,66),(2,76,40,13,67),(3,77,33,14,68),(4,78,34,15,69),(5,79,35,16,70),(6,80,36,9,71),(7,73,37,10,72),(8,74,38,11,65),(17,43,52,64,26),(18,44,53,57,27),(19,45,54,58,28),(20,46,55,59,29),(21,47,56,60,30),(22,48,49,61,31),(23,41,50,62,32),(24,42,51,63,25)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,65),(9,80),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,41),(49,61),(50,62),(51,63),(52,64),(53,57),(54,58),(55,59),(56,60)], [(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)], [(2,8),(3,7),(4,6),(9,78,36,69),(10,77,37,68),(11,76,38,67),(12,75,39,66),(13,74,40,65),(14,73,33,72),(15,80,34,71),(16,79,35,70),(17,50,43,32),(18,49,44,31),(19,56,45,30),(20,55,46,29),(21,54,47,28),(22,53,48,27),(23,52,41,26),(24,51,42,25),(57,61),(58,60),(62,64)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 10 | 10 | 20 | ··· | 20 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | - | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | Q8 | D4 | D8 | Q16 | F5 | C2×F5 | C2×F5 | C4⋊F5 | C4⋊F5 | D5.D8 |
| kernel | C2×D5.D8 | D5.D8 | D5×C2×C8 | C2×C4⋊F5 | C8×D5 | C2×C5⋊2C8 | C2×C40 | C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | D10 | D10 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 8 |
Matrix representation of C2×D5.D8 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 35 | 31 | 0 | 0 | 0 | 0 |
| 16 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 37 | 37 |
| 0 | 0 | 4 | 31 | 4 | 0 |
| 0 | 0 | 0 | 4 | 31 | 4 |
| 0 | 0 | 37 | 37 | 0 | 27 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 22 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 40 | 40 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0,0,40,0,1,0,0,0,40,1,0,0],[35,16,0,0,0,0,31,6,0,0,0,0,0,0,27,4,0,37,0,0,0,31,4,37,0,0,37,4,31,0,0,0,37,0,4,27],[9,22,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;
C2×D5.D8 in GAP, Magma, Sage, TeX
C_2\times D_5.D_8
% in TeX
G:=Group("C2xD5.D8"); // GroupNames label
G:=SmallGroup(320,1058);
// by ID
G=gap.SmallGroup(320,1058);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,268,1684,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^8=1,e^2=b^-1*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^3,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=d^-1>;
// generators/relations