direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D5×D8, C40⋊3C23, D20⋊1C23, C20.1C24, D40⋊15C22, C10⋊2(C2×D8), C5⋊2(C22×D8), (C10×D8)⋊7C2, (C2×C8)⋊26D10, C4.39(D4×D5), C8⋊5(C22×D5), (C2×D40)⋊19C2, (C2×D4)⋊27D10, C5⋊2C8⋊6C23, D4⋊D5⋊8C22, (C4×D5).65D4, C20.76(C2×D4), D4⋊1(C22×D5), (C5×D8)⋊9C22, (C5×D4)⋊1C23, (D4×D5)⋊4C22, C4.1(C23×D5), (C2×C40)⋊11C22, (C8×D5)⋊13C22, D10.111(C2×D4), (D4×C10)⋊18C22, (C2×D20)⋊32C22, Dic5.23(C2×D4), (C4×D5).59C23, C22.135(D4×D5), (C2×C20).518C23, (C2×Dic5).166D4, (C22×D5).158D4, C10.102(C22×D4), (D5×C2×C8)⋊4C2, (C2×D4×D5)⋊21C2, C2.75(C2×D4×D5), (C2×D4⋊D5)⋊25C2, (C2×C5⋊2C8)⋊35C22, (C2×C10).391(C2×D4), (C2×C4×D5).325C22, (C2×C4).608(C22×D5), SmallGroup(320,1426)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D5×D8
G = < a,b,c,d,e | a2=b5=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1694 in 338 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, C23, D5, D5, C10, C10, C10, C2×C8, C2×C8, D8, D8, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C22×C8, C2×D8, C2×D8, C22×D4, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×D4, C22×D5, C22×D5, C22×C10, C22×D8, C8×D5, D40, C2×C5⋊2C8, D4⋊D5, C2×C40, C5×D8, C2×C4×D5, C2×D20, D4×D5, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, D5×C2×C8, C2×D40, D5×D8, C2×D4⋊D5, C10×D8, C2×D4×D5, C2×D5×D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C24, D10, C2×D8, C22×D4, C22×D5, C22×D8, D4×D5, C23×D5, D5×D8, C2×D4×D5, C2×D5×D8
►On 80 points
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 65)(16 66)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 49)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(41 62)(42 63)(43 64)(44 57)(45 58)(46 59)(47 60)(48 61)
(1 59 17 25 67)(2 60 18 26 68)(3 61 19 27 69)(4 62 20 28 70)(5 63 21 29 71)(6 64 22 30 72)(7 57 23 31 65)(8 58 24 32 66)(9 39 46 50 73)(10 40 47 51 74)(11 33 48 52 75)(12 34 41 53 76)(13 35 42 54 77)(14 36 43 55 78)(15 37 44 56 79)(16 38 45 49 80)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 65)(8 66)(9 39)(10 40)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(41 76)(42 77)(43 78)(44 79)(45 80)(46 73)(47 74)(48 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)(10 16)(11 15)(12 14)(18 24)(19 23)(20 22)(26 32)(27 31)(28 30)(33 37)(34 36)(38 40)(41 43)(44 48)(45 47)(49 51)(52 56)(53 55)(57 61)(58 60)(62 64)(65 69)(66 68)(70 72)(74 80)(75 79)(76 78)
G:=sub<Sym(80)| (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,65)(16,66)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,49)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(41,62)(42,63)(43,64)(44,57)(45,58)(46,59)(47,60)(48,61), (1,59,17,25,67)(2,60,18,26,68)(3,61,19,27,69)(4,62,20,28,70)(5,63,21,29,71)(6,64,22,30,72)(7,57,23,31,65)(8,58,24,32,66)(9,39,46,50,73)(10,40,47,51,74)(11,33,48,52,75)(12,34,41,53,76)(13,35,42,54,77)(14,36,43,55,78)(15,37,44,56,79)(16,38,45,49,80), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,65)(8,66)(9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(41,76)(42,77)(43,78)(44,79)(45,80)(46,73)(47,74)(48,75), (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)(10,16)(11,15)(12,14)(18,24)(19,23)(20,22)(26,32)(27,31)(28,30)(33,37)(34,36)(38,40)(41,43)(44,48)(45,47)(49,51)(52,56)(53,55)(57,61)(58,60)(62,64)(65,69)(66,68)(70,72)(74,80)(75,79)(76,78)>;
G:=Group( (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,65)(16,66)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,49)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(41,62)(42,63)(43,64)(44,57)(45,58)(46,59)(47,60)(48,61), (1,59,17,25,67)(2,60,18,26,68)(3,61,19,27,69)(4,62,20,28,70)(5,63,21,29,71)(6,64,22,30,72)(7,57,23,31,65)(8,58,24,32,66)(9,39,46,50,73)(10,40,47,51,74)(11,33,48,52,75)(12,34,41,53,76)(13,35,42,54,77)(14,36,43,55,78)(15,37,44,56,79)(16,38,45,49,80), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,65)(8,66)(9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(41,76)(42,77)(43,78)(44,79)(45,80)(46,73)(47,74)(48,75), (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)(10,16)(11,15)(12,14)(18,24)(19,23)(20,22)(26,32)(27,31)(28,30)(33,37)(34,36)(38,40)(41,43)(44,48)(45,47)(49,51)(52,56)(53,55)(57,61)(58,60)(62,64)(65,69)(66,68)(70,72)(74,80)(75,79)(76,78) );
G=PermutationGroup([[(1,39),(2,40),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,65),(16,66),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,49),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(41,62),(42,63),(43,64),(44,57),(45,58),(46,59),(47,60),(48,61)], [(1,59,17,25,67),(2,60,18,26,68),(3,61,19,27,69),(4,62,20,28,70),(5,63,21,29,71),(6,64,22,30,72),(7,57,23,31,65),(8,58,24,32,66),(9,39,46,50,73),(10,40,47,51,74),(11,33,48,52,75),(12,34,41,53,76),(13,35,42,54,77),(14,36,43,55,78),(15,37,44,56,79),(16,38,45,49,80)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,65),(8,66),(9,39),(10,40),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,57),(32,58),(41,76),(42,77),(43,78),(44,79),(45,80),(46,73),(47,74),(48,75)], [(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),(10,16),(11,15),(12,14),(18,24),(19,23),(20,22),(26,32),(27,31),(28,30),(33,37),(34,36),(38,40),(41,43),(44,48),(45,47),(49,51),(52,56),(53,55),(57,61),(58,60),(62,64),(65,69),(66,68),(70,72),(74,80),(75,79),(76,78)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 20 | 20 | 20 | 20 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D8 | D10 | D10 | D10 | D4×D5 | D4×D5 | D5×D8 |
| kernel | C2×D5×D8 | D5×C2×C8 | C2×D40 | D5×D8 | C2×D4⋊D5 | C10×D8 | C2×D4×D5 | C4×D5 | C2×Dic5 | C22×D5 | C2×D8 | D10 | C2×C8 | D8 | C2×D4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 8 | 2 | 8 | 4 | 2 | 2 | 8 |
Matrix representation of C2×D5×D8 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 40 | 34 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 17 | 24 |
| 0 | 0 | 29 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,1,34,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,17,29,0,0,24,0],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,40] >;
C2×D5×D8 in GAP, Magma, Sage, TeX
C_2\times D_5\times D_8
% in TeX
G:=Group("C2xD5xD8"); // GroupNames label
G:=SmallGroup(320,1426);
// by ID
G=gap.SmallGroup(320,1426);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,185,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^8=e^2=1,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,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations