direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.46D4, M4(2)⋊22D10, C4.65(C2×D20), (C2×C4).49D20, (C2×D20).27C4, C20.416(C2×D4), (C2×C20).172D4, (C23×D5).3C4, C23.55(C4×D5), C10⋊2(C4.D4), (C2×M4(2))⋊10D5, (C10×M4(2))⋊18C2, (C2×C20).416C23, (C22×D20).15C2, (C22×C4).138D10, C4.Dic5⋊21C22, C4.28(D10⋊C4), C20.100(C22⋊C4), (C2×D20).258C22, (C5×M4(2))⋊34C22, (C22×C20).187C22, C22.50(D10⋊C4), C5⋊4(C2×C4.D4), (C2×C4).52(C4×D5), C22.20(C2×C4×D5), C4.109(C2×C5⋊D4), (C2×C20).280(C2×C4), C10.98(C2×C22⋊C4), (C2×C4.Dic5)⋊15C2, (C22×D5).5(C2×C4), C2.29(C2×D10⋊C4), (C2×C4).256(C5⋊D4), (C2×C4).120(C22×D5), (C22×C10).138(C2×C4), (C2×C10).115(C22×C4), (C2×C10).129(C22⋊C4), SmallGroup(320,757)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C20.46D4
G = < a,b,c,d | a2=b20=d2=1, c4=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b15c3 >
Subgroups: 958 in 186 conjugacy classes, 63 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C22 [×3], C22 [×18], C5, C8 [×4], C2×C4 [×6], D4 [×8], C23, C23 [×12], D5 [×4], C10, C10 [×2], C10 [×2], C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4, C2×D4 [×8], C24 [×2], C20 [×4], D10 [×16], C2×C10 [×3], C2×C10 [×2], C4.D4 [×4], C2×M4(2), C2×M4(2), C22×D4, C5⋊2C8 [×2], C40 [×2], D20 [×8], C2×C20 [×6], C22×D5 [×4], C22×D5 [×8], C22×C10, C2×C4.D4, C2×C5⋊2C8, C4.Dic5 [×2], C4.Dic5, C2×C40, C5×M4(2) [×2], C5×M4(2), C2×D20 [×4], C2×D20 [×4], C22×C20, C23×D5 [×2], C20.46D4 [×4], C2×C4.Dic5, C10×M4(2), C22×D20, C2×C20.46D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C4.D4 [×2], C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C2×C4.D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C20.46D4 [×2], C2×D10⋊C4, C2×C20.46D4
►On 80 points
(1 78)(2 79)(3 80)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 73)(17 74)(18 75)(19 76)(20 77)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(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 48 73 25 11 58 63 35)(2 47 74 24 12 57 64 34)(3 46 75 23 13 56 65 33)(4 45 76 22 14 55 66 32)(5 44 77 21 15 54 67 31)(6 43 78 40 16 53 68 30)(7 42 79 39 17 52 69 29)(8 41 80 38 18 51 70 28)(9 60 61 37 19 50 71 27)(10 59 62 36 20 49 72 26)
(1 73)(2 72)(3 71)(4 70)(5 69)(6 68)(7 67)(8 66)(9 65)(10 64)(11 63)(12 62)(13 61)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)
G:=sub<Sym(80)| (1,78)(2,79)(3,80)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,73)(17,74)(18,75)(19,76)(20,77)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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), (1,48,73,25,11,58,63,35)(2,47,74,24,12,57,64,34)(3,46,75,23,13,56,65,33)(4,45,76,22,14,55,66,32)(5,44,77,21,15,54,67,31)(6,43,78,40,16,53,68,30)(7,42,79,39,17,52,69,29)(8,41,80,38,18,51,70,28)(9,60,61,37,19,50,71,27)(10,59,62,36,20,49,72,26), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)>;
G:=Group( (1,78)(2,79)(3,80)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,73)(17,74)(18,75)(19,76)(20,77)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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), (1,48,73,25,11,58,63,35)(2,47,74,24,12,57,64,34)(3,46,75,23,13,56,65,33)(4,45,76,22,14,55,66,32)(5,44,77,21,15,54,67,31)(6,43,78,40,16,53,68,30)(7,42,79,39,17,52,69,29)(8,41,80,38,18,51,70,28)(9,60,61,37,19,50,71,27)(10,59,62,36,20,49,72,26), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56) );
G=PermutationGroup([(1,78),(2,79),(3,80),(4,61),(5,62),(6,63),(7,64),(8,65),(9,66),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,73),(17,74),(18,75),(19,76),(20,77),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(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)], [(1,48,73,25,11,58,63,35),(2,47,74,24,12,57,64,34),(3,46,75,23,13,56,65,33),(4,45,76,22,14,55,66,32),(5,44,77,21,15,54,67,31),(6,43,78,40,16,53,68,30),(7,42,79,39,17,52,69,29),(8,41,80,38,18,51,70,28),(9,60,61,37,19,50,71,27),(10,59,62,36,20,49,72,26)], [(1,73),(2,72),(3,71),(4,70),(5,69),(6,68),(7,67),(8,66),(9,65),(10,64),(11,63),(12,62),(13,61),(14,80),(15,79),(16,78),(17,77),(18,76),(19,75),(20,74),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,40),(36,39),(37,38),(41,50),(42,49),(43,48),(44,47),(45,46),(51,60),(52,59),(53,58),(54,57),(55,56)])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 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 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 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 | C4 | C4 | D4 | D5 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | C4×D5 | C4.D4 | C20.46D4 |
| kernel | C2×C20.46D4 | C20.46D4 | C2×C4.Dic5 | C10×M4(2) | C22×D20 | C2×D20 | C23×D5 | C2×C20 | C2×M4(2) | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 4 | 2 | 4 | 8 | 8 | 4 | 2 | 8 |
Matrix representation of C2×C20.46D4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 6 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 30 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 11 |
| 0 | 0 | 0 | 0 | 14 | 25 |
| 6 | 35 | 0 | 0 | 0 | 0 |
| 40 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 5 | 4 | 9 |
| 0 | 0 | 32 | 39 | 13 | 28 |
| 0 | 0 | 4 | 21 | 12 | 8 |
| 0 | 0 | 38 | 17 | 23 | 29 |
| 6 | 35 | 0 | 0 | 0 | 0 |
| 40 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 14 | 30 | 0 | 0 |
| 0 | 0 | 29 | 39 | 25 | 30 |
| 0 | 0 | 29 | 10 | 12 | 16 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[6,40,0,0,0,0,1,0,0,0,0,0,0,0,9,30,0,0,0,0,11,14,0,0,0,0,0,0,39,14,0,0,0,0,11,25],[6,40,0,0,0,0,35,35,0,0,0,0,0,0,2,32,4,38,0,0,5,39,21,17,0,0,4,13,12,23,0,0,9,28,8,29],[6,40,0,0,0,0,35,35,0,0,0,0,0,0,11,14,29,29,0,0,9,30,39,10,0,0,0,0,25,12,0,0,0,0,30,16] >;
C2×C20.46D4 in GAP, Magma, Sage, TeX
C_2\times C_{20}._{46}D_4 % in TeX
G:=Group("C2xC20.46D4"); // GroupNames label
G:=SmallGroup(320,757);
// by ID
G=gap.SmallGroup(320,757);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,58,1123,136,438,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=d^2=1,c^4=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=b^15*c^3>;
// generators/relations