direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D5.D8, D10.11D8, D10.6Q16, C8⋊7(C2×F5), (C2×C8)⋊5F5, C40⋊7(C2×C4), (C2×C40)⋊5C4, (C8×D5)⋊6C4, C10⋊(C2.D8), D5⋊(C2.D8), D5.1(C2×D8), (C4×D5).83D4, C4.13(C4⋊F5), C20.20(C4⋊C4), (C4×D5).25Q8, D5.1(C2×Q16), D10.30(C2×D4), C4⋊F5.16C22, D10.28(C4⋊C4), C4.36(C22×F5), C20.76(C22×C4), Dic5.14(C2×Q8), (C2×Dic5).34Q8, (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
Subgroups: 538 in 130 conjugacy classes, 60 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×13], C23, D5 [×2], D5 [×2], C10, C10 [×2], C4⋊C4 [×6], C2×C8, C2×C8 [×5], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×4], C2×C10, C2.D8 [×4], C2×C4⋊C4 [×2], C22×C8, C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×8], C22×D5, C2×C2.D8, C8×D5 [×4], C2×C5⋊2C8, C2×C40, C4⋊F5 [×4], C4⋊F5 [×2], C2×C4×D5, C22×F5 [×2], D5.D8 [×4], D5×C2×C8, C2×C4⋊F5 [×2], C2×D5.D8
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], D8 [×2], Q16 [×2], C22×C4, C2×D4, C2×Q8, F5, C2.D8 [×4], C2×C4⋊C4, C2×D8, C2×Q16, C2×F5 [×3], C2×C2.D8, C4⋊F5 [×2], C22×F5, D5.D8 [×2], C2×C4⋊F5, C2×D5.D8
Generators and relations
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 >
►On 80 points
(1 78)(2 79)(3 80)(4 73)(5 74)(6 75)(7 76)(8 77)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)(49 72)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)
(1 15 26 62 66)(2 16 27 63 67)(3 9 28 64 68)(4 10 29 57 69)(5 11 30 58 70)(6 12 31 59 71)(7 13 32 60 72)(8 14 25 61 65)(17 38 52 79 46)(18 39 53 80 47)(19 40 54 73 48)(20 33 55 74 41)(21 34 56 75 42)(22 35 49 76 43)(23 36 50 77 44)(24 37 51 78 45)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 65)(9 64)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 76)(50 77)(51 78)(52 79)(53 80)(54 73)(55 74)(56 75)
(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 32 68 60)(10 31 69 59)(11 30 70 58)(12 29 71 57)(13 28 72 64)(14 27 65 63)(15 26 66 62)(16 25 67 61)(17 50 38 44)(18 49 39 43)(19 56 40 42)(20 55 33 41)(21 54 34 48)(22 53 35 47)(23 52 36 46)(24 51 37 45)(73 75)(76 80)(77 79)
G:=sub<Sym(80)| (1,78)(2,79)(3,80)(4,73)(5,74)(6,75)(7,76)(8,77)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57)(49,72)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71), (1,15,26,62,66)(2,16,27,63,67)(3,9,28,64,68)(4,10,29,57,69)(5,11,30,58,70)(6,12,31,59,71)(7,13,32,60,72)(8,14,25,61,65)(17,38,52,79,46)(18,39,53,80,47)(19,40,54,73,48)(20,33,55,74,41)(21,34,56,75,42)(22,35,49,76,43)(23,36,50,77,44)(24,37,51,78,45), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,65)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,76)(50,77)(51,78)(52,79)(53,80)(54,73)(55,74)(56,75), (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)(9,32,68,60)(10,31,69,59)(11,30,70,58)(12,29,71,57)(13,28,72,64)(14,27,65,63)(15,26,66,62)(16,25,67,61)(17,50,38,44)(18,49,39,43)(19,56,40,42)(20,55,33,41)(21,54,34,48)(22,53,35,47)(23,52,36,46)(24,51,37,45)(73,75)(76,80)(77,79)>;
G:=Group( (1,78)(2,79)(3,80)(4,73)(5,74)(6,75)(7,76)(8,77)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57)(49,72)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71), (1,15,26,62,66)(2,16,27,63,67)(3,9,28,64,68)(4,10,29,57,69)(5,11,30,58,70)(6,12,31,59,71)(7,13,32,60,72)(8,14,25,61,65)(17,38,52,79,46)(18,39,53,80,47)(19,40,54,73,48)(20,33,55,74,41)(21,34,56,75,42)(22,35,49,76,43)(23,36,50,77,44)(24,37,51,78,45), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,65)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,76)(50,77)(51,78)(52,79)(53,80)(54,73)(55,74)(56,75), (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)(9,32,68,60)(10,31,69,59)(11,30,70,58)(12,29,71,57)(13,28,72,64)(14,27,65,63)(15,26,66,62)(16,25,67,61)(17,50,38,44)(18,49,39,43)(19,56,40,42)(20,55,33,41)(21,54,34,48)(22,53,35,47)(23,52,36,46)(24,51,37,45)(73,75)(76,80)(77,79) );
G=PermutationGroup([(1,78),(2,79),(3,80),(4,73),(5,74),(6,75),(7,76),(8,77),(9,47),(10,48),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,25),(24,26),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,57),(49,72),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71)], [(1,15,26,62,66),(2,16,27,63,67),(3,9,28,64,68),(4,10,29,57,69),(5,11,30,58,70),(6,12,31,59,71),(7,13,32,60,72),(8,14,25,61,65),(17,38,52,79,46),(18,39,53,80,47),(19,40,54,73,48),(20,33,55,74,41),(21,34,56,75,42),(22,35,49,76,43),(23,36,50,77,44),(24,37,51,78,45)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,65),(9,64),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(49,76),(50,77),(51,78),(52,79),(53,80),(54,73),(55,74),(56,75)], [(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),(9,32,68,60),(10,31,69,59),(11,30,70,58),(12,29,71,57),(13,28,72,64),(14,27,65,63),(15,26,66,62),(16,25,67,61),(17,50,38,44),(18,49,39,43),(19,56,40,42),(20,55,33,41),(21,54,34,48),(22,53,35,47),(23,52,36,46),(24,51,37,45),(73,75),(76,80),(77,79)])
Matrix representation ►G ⊆ 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] >;
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 |
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