direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C8⋊D10, C40⋊2C23, D40⋊9C22, D20⋊5C23, C20.58C24, C23.52D20, M4(2)⋊17D10, Dic10⋊5C23, (C2×C8)⋊4D10, C8⋊2(C22×D5), (C2×D40)⋊14C2, (C2×C40)⋊7C22, (C2×C4).57D20, C4.48(C2×D20), C10⋊1(C8⋊C22), C20.238(C2×D4), (C2×C20).203D4, C40⋊C2⋊8C22, (C2×M4(2))⋊3D5, C4.55(C23×D5), C4○D20⋊18C22, (C22×D20)⋊17C2, (C2×D20)⋊49C22, (C10×M4(2))⋊3C2, C2.27(C22×D20), C22.73(C2×D20), C10.25(C22×D4), (C2×C20).511C23, (C22×C10).118D4, (C22×C4).265D10, (C2×Dic10)⋊57C22, (C5×M4(2))⋊19C22, (C22×C20).266C22, C5⋊1(C2×C8⋊C22), (C2×C40⋊C2)⋊4C2, (C2×C4○D20)⋊26C2, (C2×C10).62(C2×D4), (C2×C4).223(C22×D5), SmallGroup(320,1418)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1438 in 298 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×17], Q8 [×3], C23, C23 [×11], D5 [×6], C10, C10 [×2], C10 [×2], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4 [×11], C2×Q8, C4○D4 [×6], C24, Dic5 [×2], C20 [×2], C20 [×2], D10 [×20], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C40 [×4], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×6], D20 [×7], C2×Dic5, C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C22×D5 [×11], C22×C10, C2×C8⋊C22, C40⋊C2 [×8], D40 [×8], C2×C40 [×2], C5×M4(2) [×4], C2×Dic10, C2×C4×D5, C2×D20, C2×D20 [×6], C2×D20 [×3], C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, C23×D5, C2×C40⋊C2 [×2], C2×D40 [×2], C8⋊D10 [×8], C10×M4(2), C22×D20, C2×C4○D20, C2×C8⋊D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8⋊C22 [×2], C22×D4, D20 [×4], C22×D5 [×7], C2×C8⋊C22, C2×D20 [×6], C23×D5, C8⋊D10 [×2], C22×D20, C2×C8⋊D10
Generators and relations
G = < a,b,c,d | a2=b8=c10=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >
►On 80 points
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 41)(31 36)(32 37)(33 38)(34 39)(35 40)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 66)(62 67)(63 68)(64 69)(65 70)
(1 25 40 52 67 41 19 77)(2 42 31 78 68 26 20 53)(3 27 32 54 69 43 11 79)(4 44 33 80 70 28 12 55)(5 29 34 56 61 45 13 71)(6 46 35 72 62 30 14 57)(7 21 36 58 63 47 15 73)(8 48 37 74 64 22 16 59)(9 23 38 60 65 49 17 75)(10 50 39 76 66 24 18 51)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 40)(19 39)(20 38)(21 55)(22 54)(23 53)(24 52)(25 51)(26 60)(27 59)(28 58)(29 57)(30 56)(41 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 80)(48 79)(49 78)(50 77)(61 62)(63 70)(64 69)(65 68)(66 67)
G:=sub<Sym(80)| (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,41)(31,36)(32,37)(33,38)(34,39)(35,40)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,66)(62,67)(63,68)(64,69)(65,70), (1,25,40,52,67,41,19,77)(2,42,31,78,68,26,20,53)(3,27,32,54,69,43,11,79)(4,44,33,80,70,28,12,55)(5,29,34,56,61,45,13,71)(6,46,35,72,62,30,14,57)(7,21,36,58,63,47,15,73)(8,48,37,74,64,22,16,59)(9,23,38,60,65,49,17,75)(10,50,39,76,66,24,18,51), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,40)(19,39)(20,38)(21,55)(22,54)(23,53)(24,52)(25,51)(26,60)(27,59)(28,58)(29,57)(30,56)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,80)(48,79)(49,78)(50,77)(61,62)(63,70)(64,69)(65,68)(66,67)>;
G:=Group( (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,41)(31,36)(32,37)(33,38)(34,39)(35,40)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,66)(62,67)(63,68)(64,69)(65,70), (1,25,40,52,67,41,19,77)(2,42,31,78,68,26,20,53)(3,27,32,54,69,43,11,79)(4,44,33,80,70,28,12,55)(5,29,34,56,61,45,13,71)(6,46,35,72,62,30,14,57)(7,21,36,58,63,47,15,73)(8,48,37,74,64,22,16,59)(9,23,38,60,65,49,17,75)(10,50,39,76,66,24,18,51), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,40)(19,39)(20,38)(21,55)(22,54)(23,53)(24,52)(25,51)(26,60)(27,59)(28,58)(29,57)(30,56)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,80)(48,79)(49,78)(50,77)(61,62)(63,70)(64,69)(65,68)(66,67) );
G=PermutationGroup([(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,50),(30,41),(31,36),(32,37),(33,38),(34,39),(35,40),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,66),(62,67),(63,68),(64,69),(65,70)], [(1,25,40,52,67,41,19,77),(2,42,31,78,68,26,20,53),(3,27,32,54,69,43,11,79),(4,44,33,80,70,28,12,55),(5,29,34,56,61,45,13,71),(6,46,35,72,62,30,14,57),(7,21,36,58,63,47,15,73),(8,48,37,74,64,22,16,59),(9,23,38,60,65,49,17,75),(10,50,39,76,66,24,18,51)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,37),(12,36),(13,35),(14,34),(15,33),(16,32),(17,31),(18,40),(19,39),(20,38),(21,55),(22,54),(23,53),(24,52),(25,51),(26,60),(27,59),(28,58),(29,57),(30,56),(41,76),(42,75),(43,74),(44,73),(45,72),(46,71),(47,80),(48,79),(49,78),(50,77),(61,62),(63,70),(64,69),(65,68),(66,67)])
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 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 7 |
| 0 | 0 | 0 | 0 | 34 | 34 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 14 |
| 0 | 0 | 0 | 0 | 9 | 11 |
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],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,7,0,0,0,0,34,7,0,0,0,0,0,0,1,34,0,0,0,0,7,34],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,34,1,0,0,0,0,0,0,30,9,0,0,0,0,14,11] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | ··· | 20 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D20 | D20 | C8⋊C22 | C8⋊D10 |
| kernel | C2×C8⋊D10 | C2×C40⋊C2 | C2×D40 | C8⋊D10 | C10×M4(2) | C22×D20 | C2×C4○D20 | C2×C20 | C22×C10 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 3 | 1 | 2 | 4 | 8 | 2 | 12 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_8\rtimes D_{10} % in TeX
G:=Group("C2xC8:D10"); // GroupNames label
G:=SmallGroup(320,1418);
// by ID
G=gap.SmallGroup(320,1418);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,675,297,80,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^10=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations